Basalt by definition, according to the International Union of Geological Sciences classification system, is a fine-grained igneous rock with somewhere between 45% and 53% silica (SiO2) and less than 10% feldspathoid (very similar to feldspar but with a different structure and lower silica levels)  by volume, and where at least 65% of the rock is feldspar in the form of plagioclase[1]. It is the most common volcanic rock type found on Earth and formed from the rapid cooling of magnesium-rich and iron-rich lava due to volcanic eruptions. It is a key component of the Earth’s crust, making up the majority of the oceanic floor and many the mid-oceanic islands, including Iceland, the Faroe Islands, Réunion and the islands of Hawaiʻi.

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So, where does basalt come from? Most of the basalt found on Earth has been produced either by oceanic divergent boundaries, oceanic hotspots, or mantle plumes and hotspots underneath continents. The lava when it reaches the surface, from the mantle, is about 1100 to 1250° C. It then cools over a span of a few days or weeks and forms solid igneous rock. The two types of volcanic basalt are described by the Hawaiian words ‘a’a and pahoehoe. ‘A’a basalts have rough, almost pointed, jagged surfaces, and form from fast flowing lava. While pahoehoe basalts have a smooth and odd rope like texture. The ropes or some even say waves in the rock form when the surface of the flow cools, but the lava underneath continues to move[2]. These basalts on earth have a pretty consistent make up, usually having a composition of of 45–55 wt% SiO2 (as stated above), 2–6 wt% total alkalis, 0.5–2.0 wt% TiO2, 5–14 wt% FeO and 14 wt% or more Al2O3. There is also the amounts of CaO that are commonly near 10 wt%, and of MgO which can possibly range from 5 to 12 wt%[3]. These compositions change however when one moves slightly away from our terrestrial body these compositions seem to differ.  
The closest terrestrial body to us is the moon. Now there are many hypotheses about how the moon was formed such as it is believed that the moon is essentially just a chunk of the earth that was thrown into space during a very cataclysmic event early in Earth’s history. Whatever the case may have been, the lunar basaltic compositions, although similar, differ from Earth’s is several ways. Most basalt knowledge from the moon comes from studying lunar maria which are dark, flat, and often circular regions as seen on the moon. The formation of these mare basalts was covered in Taylor (2007) and it is said that they likely originated through partial melting at depths around 300 km in the lunar interior and at temperatures of about 1200°C. The basalts are derived from the zones and mounds of minerals developed at varying depths during crystallization of the magma ocean that the moon once had. The isotopic make up of these mare basalts seem to indicate that the source region had crystallized by approximately 4.4 billion years. Partial melting then occurred hundreds of millions of years later in these mineral zones due to the buildup of heat caused by radioactive elements. Around 25 types of mare basalt were erupted over an interval of more than 1 billion years, but the total amount of melt so generated amounted to only about 0.1% of the volume of the Moon. This forms a stark contrast to the state of the Moon at accretion, when it may have been entirely molten[4].
Lunar basalt samples seem to differ most noticeably in their iron contents, which is slightly higher and usually range from about 17 to 22 wt.% FeO[5]. Lunar basalts also contain a wide range of titanium concentrations found in ilmenite, ranging from less than 1 wt.% TiO2, to up to around 13 wt.%[6]. This has led to most lunar basalts being classified according to their titanium content, with classes being high-titanium, low-titanium, and very-low-titanium. Global geochemical maps of titanium acquired from the Clementine mission demonstrated that the lunar maria possess a wide range of titanium concentrations, and that the highest concentrations are the least abundant. These varying concentrations most likely reflect the relative abundances and lack of abundance of ilmenite mantle sources in certain areas. This theory is backed by the distribution being consistent with the models of the formation of mare source regions from the lunar magma ocean5.
The second terrestrial body that is known for its basalts is in fact another planet, Mars. As mentioned above the most common form of volcanism on the Earth is basaltic, and this is most likely the truth when it comes to Mars as well. One key difference, however, between the two bodies is their slight differences in formations due to the differing environments. On Earth, magma that forms basalts usually erupts as highly fluid flows, which can emerge either directly from vents or other sources already stated. Although these styles are also common on Mars, the lower gravity and lower atmospheric pressure on the planet allows formation of gas bubbles to occur more frequently and at greater depths than on our planet. As a result, Martian basaltic volcanoes are also able to have Plinian or Vesuvian-style eruptions and throw out large quantities of ash. These eruptions of course being named after the infamous eruption of Mount Vesuvius. The lower gravity of Mars also generates less buoyancy forces on magma rising through the crust, therefore if magma on Mars is able to climb and get close enough to the surface to erupt before solidifying, it is most likely quite a large body. This in turn means that eruptions on Mars are less frequent than on Earth but are on a much more enormous in scale and eruptive rate[7]. In somewhat puzzling fashion however, the lower gravity of Mars also allows for longer and more widespread lava flows. Lava eruptions on Mars have the potential to be unimaginably huge, such as one the size of the state of Oregon that has been recently has discovered in western Elysium Planitia[8]. The flow is believed to be one of the youngest lava flows on Mars and happened over the course of several weeks.
With only slight differences in the formation processes of Martian basalt and Earth basalts, it is no surprise that the two terrestrial bodies have very similar chemical compositions when it comes to their igneous rocks. It has been determined that the dominant surface rocks on Mars are tholeiitic basalts that were most likely formed by partial melting and do not show signs of any extreme weathering. In October of 2012, the Curiosity rover at the Rocknest site on Mars performed the first diffraction analysis of Martian soil and chemically revealed the presence of several minerals all usually present is basalts. This included feldspar, pyroxenes and olivine, and it is said that the Martian soil sample from this particular area had characteristics similar to weathered basaltic rocks of the Hawaiian Islands[9].
The chemical composition of these basalts has been studied through the testing of shergottite meteorite (named after the Shergotty meteorite) basalts. These tests can and have provided important knowledge on magma origin and mantle processes in Mars and were highlighted by Trieman (2003). From these analyses of the Martian meteorites two separate groups were created. These two groups are aptly named Group 1 (Gl), which includes highly incompatible elements such as La and Th and Group 2 (G2), which includes moderately incompatible elements such as Ti, Lu, and Al. Correlated variations of these G2 is consistent with partitioning between basalt magma and pyroxene and olivine. This fractionation is a result of partial melting to form the shergottites and their crystallization. All in all, abundances of Gl elements are separate from those of G2. When comparing abundances of Gl elements with the abundances of G2 elements, the ratios do not appear to be random, however, shergottites with a certain ratio do not necessarily have the same crystallization age and may also not fall on a single fractionation trajectory. These observations point to the G1/G2 families being established before basalt formation. It also suggests enrichment of their source region of high G2 elements, by a GI rich component. It would seem that Group 1 elements were efficiently separated from G2 elements very early in Mars’ history. The efficiency at which the fractionation seemed to have occurred is not consistent with simple petrogenesis; it requires many fractionations, and a more complex process[10]. The behavior of phosphorus in these early fractionation events is unheard of and hard to explain by normal processes and minerals. Several explanations have been brought up, however, and are possible.
As one can see, basalt seems to be a consistent entity on several terrestrial bodies in space. Although I did not delve into other bodies, they do include the asteroid Vesta and other terrestrial planets where basalt has been found. These basalts can tell us a lot about formation and mantle/magma processes of terrestrial bodies, yet there is still much to learn. I feel that the desire to find similarities between earth and other bodies gives us hope and the idea that there is the possibility that we will find a similar planet to ours.
References

Bas, M. J. Le, and A. L. Streckeisen. “The IUGS Systematics of Igneous Rocks.” Journal of the Geological Society, vol. 148, no. 5, 1991, pp. 825–833. GeoScienceWorld, doi:10.1144/gsjgs.148.5.0825.
“Basalt.” Wikipedia, Wikimedia Foundation, 11 Dec. 2019, https://en.wikipedia.org/wiki/Basalt#Morphology_and_textures.
“Basalt Rocks.” Windows to the Universe, 1 Nov. 2005, https://www.windows2universe.org/earth/geology/ig_basalt.html.
Giguere, Thomas A., et al. “The Titanium Contents of Lunar Mare Basalts.” Meteoritics & Planetary Science, vol. 35, no. 1, 4 Feb. 2000, pp. 193–200. Wiley Online Library, doi:10.1111/j.1945-5100.2000.tb01985.x.
Grotzinger, J. P. “Analysis of Surface Materials by the Curiosity Mars Rover.” Science, vol. 341, no. 6153, 2013, pp. 1475–1475., doi:10.1126/science.1244258.
Jaeger, W.l., et al. “Emplacement of the Youngest Flood Lava on Mars: A Short, Turbulent Story.” Icarus, vol. 205, no. 1, Jan. 2010, pp. 230–243., doi:10.1016/j.icarus.2009.09.011.
Ling, Zongcheng, et al. “Correlated Compositional and Mineralogical Investigations at the Chang′e-3 Landing Site.” Nature Communications, vol. 6, no. 1, 22 Dec. 2015, doi:10.1038/ncomms9880.
Mcsween, H. Y., et al. “Elemental Composition of the Martian Crust.” Science, vol. 324, no. 5928, 7 May 2009, pp. 736–739., doi:10.1126/science.1165871.
Program, Volcano Hazards. “Basalts.” USGS, 8 Apr. 2015, https://volcanoes.usgs.gov/vsc/glossary/basalt.html.
“NASA.gov.” NASA.gov, 30 Oct. 2012, https://www.nasa.gov/home/hqnews/2012/oct/HQ_12-383_Curiosity_CheMin.html.
Taylor, Stuart Ross. “The Moon.” Encyclopedia of the Solar System, 2007, pp. 227–250. Science Direct, doi:10.1016/b978-012088589-3/50016-5.
Treiman, Allan H. “Chemical Compositions of Martian Basalts (Shergottites): Some Inferences on b; Formation, Mantle Metasomatism, and Differentiation in Mars.” Meteoritics & Planetary Science, vol. 38, no. 12, 2003, pp. 1849–1864. Wiley Online Library, doi:10.1111/j.1945-5100.2003.tb00019.x.
Vickers, Les, et al. Fire-Resistant Geopolymers: Role of Fibres and Fillers to Enhance Thermal Properties. Springer, 2015.
Wilson, Lionel, and James W. Head. “Mars: Review and Analysis of Volcanic Eruption Theory and Relationships to Observed Landforms.” Reviews of Geophysics, vol. 32, no. 3, Aug. 1994, pp. 221–263. AGU100, doi:10.1029/94rg01113.

[1] Bas, M. J. Le, and A. L. Streckeisen. “The IUGS Systematics of Igneous Rocks.” Journal of the Geological Society, vol. 148, no. 5, 1991, pp. 825–833. GeoScienceWorld, doi:10.1144/gsjgs.148.5.0825.
[2]“Basalt Rocks.” Windows to the Universe, 1 Nov. 2005, https://www.windows2universe.org/earth/geology/ig_basalt.html.
[3]Vickers, Les, et al. Fire-Resistant Geopolymers: Role of Fibres and Fillers to Enhance Thermal Properties. Springer, 2015.
[4] Taylor, Stuart Ross. “The Moon.” Encyclopedia of the Solar System, 2007, pp. 227–250. Science Direct, doi:10.1016/b978-012088589-3/50016-5.
[5]Ling, Zongcheng, et al. “Correlated Compositional and Mineralogical Investigations at the Chang′e-3 Landing Site.” Nature Communications, vol. 6, no. 1, 22 Dec. 2015, doi:10.1038/ncomms9880.
[6] Giguere, Thomas A., et al. “The Titanium Contents of Lunar Mare Basalts.” Meteoritics & Planetary Science, vol. 35, no. 1, 4 Feb. 2000, pp. 193–200. Wiley Online Library, doi:10.1111/j.1945-5100.2000.tb01985.x.
[7]Wilson, Lionel, and James W. Head. “Mars: Review and Analysis of Volcanic Eruption Theory and Relationships to Observed Landforms.” Reviews of Geophysics, vol. 32, no. 3, Aug. 1994, pp. 221–263. AGU100, doi:10.1029/94rg01113.
[8] Jaeger, W.l., et al. “Emplacement of the Youngest Flood Lava on Mars: A Short, Turbulent Story.” Icarus, vol. 205, no. 1, Jan. 2010, pp. 230–243., doi:10.1016/j.icarus.2009.09.011.
[9] “NASA.gov.” NASA.gov, 30 Oct. 2012, https://www.nasa.gov/home/hqnews/2012/oct/HQ_12-383_Curiosity_CheMin.html.
[10] Treiman, Allan H. “Chemical Compositions of Martian Basalts (Shergottites): Some Inferences on b; Formation, Mantle Metasomatism, and Differentiation in Mars.” Meteoritics & Planetary Science, vol. 38, no. 12, 2003, pp. 1849–1864. Wiley Online Library, doi:10.1111/j.1945-5100.2003.tb00019.x.