Agile methodology is an approach to project management, typically used in software development. It refers to a group of software development methodologies based on iterative development. Requirements and solutions evolve through cooperation between self-organizing cross-functional teams, without concern for any hierarchy or team member roles. It promotes teamwork, collaboration, and process adaptability throughout the project life-cycle with increased face-to-face communication and reduced amount of written documentation.

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Agile methods break tasks into small increments with no direct long term planning. Every aspect of development is continually revisited throughout the lifecycle of project by way of iterations (also called sprints). Iterations are short time frames (“timeboxes”) that normally last 1-4 weeks. This “inspect-and-adapt” approach significantly reduces both development costs and time to market. Each iteration involves working through a complete software development cycle characterized by planning, requirements analysis, design, coding, unit testing, and acceptance testing. This helps minimize overall risk, and quicker project adaptability. While iteration may not have enough functionality necessary for a market release, aim is to be ready with a release (with minimal bugs) at the end of each iteration.
Typically, the team size is small (5-9 people) to enable easier communication and collaboration. Multiple teams may be required for larger developmental efforts which may also require a coordination of priorities across teams. Agile methods emphasize more face-to-face communication than written documents when the team is in the same location. However, when a team works at different locations, daily contact is maintained through videoconferencing, e-mail, etc. The progress made in terms of the work done today, work scheduled for tomorrow and the possible roadblocks are discussed among the team members in brief sessions at the end of the each working day. Besides, agile developmental efforts are supervised by a customer representative to ensure alignment between customer needs and company goals.
Software Development was initially based on coding and fixing. That worked well for smaller software, but as the size and complexities of software grew a need for a proper process was felt because the debugging and testing of such software became extremely difficult. This gave birth to the Engineering Methodologies.
The methodologies became highly successful since it structured the software development process. One of the most popular models that emerged was the Software Development Life Cycle (SDLC) that developed information systems in a very methodical manner.
Waterfall method is one of the most popular examples of Engineering or the SDLC methodology. A paper published by Winston Royce in 1970 introduced it as an idea. It was derived from the hardware manufacture and construction strategies that were in practice during the 1970s.
The relationship of each stage to the others can be roughly described as a waterfall, where the outputs from a specific stage serve as the initial inputs for the following stage. During each stage, additional information is gathered or developed, combined with the inputs, and used to produce the stage deliverables. It is important to note that the additional information is restricted in scope; “new ideas” that would take the project in directions not anticipated by the initial set of high-level requirements are not incorporated into the project. Rather, ideas for new capabilities or features that are out-of-scope are preserved for later consideration.
Agile Methodology:
Engineering methodologies required a lot of documentation thereby causing the pace of development to slow down considerably. Agile Methodologies evolved in the 1990s to significantly eliminate this bureaucratic nature of engineering methodology. It was part of developer’s reaction against “heavyweight” methods, who desired to drift away from traditional structured, bureaucratic approaches to software development and move towards more flexible development styles. They were called the ‘Agile’ or ‘Light Weight’ methods and were defined in 1974 by Edmonds in a research paper.
Some of the most popular agile methods that subsequently evolved were ‘Scrum’ in 1995, ‘Extreme Programming’ in 1996, ‘Adaptive Software Development’, ‘Dynamic Systems Development Method’ and ‘Feature Driven Development’. In 2001, a team of 17 pioneers in agile software development coined the terms “Agile Software development” and “agile methods”. An ‘Agile Manifesto’ was declared that was regarded as a set of canonical rules for agile software development methods and accompanying principles. Similarly, some of these people formed “The Agile Alliance’, a non-profit organization that promotes agile development.

Extreme Programming:

One of the several popular agile processes which stresses on customer satisfaction is Extreme programming. It improves a software project by focusing on communication, feedback, simplicity, courage and respect. Empowerment of the developers ensures speedier responses to changing customer requirements. Teamwork and collaboration is an integral part of extreme programming that improves the productivity and efficiency of the problem solving approach.

Comparing Agile With Waterfall Model;

Agile methodology can be compared and contrasted with traditional Waterfall model under the following heads.

Conceptual Difference: Sequential Vs Iterative:
Waterfall method is a sequential process of software development. Similar to a waterfall, the development team in such a model proceeds to the next stage of development once the first step is entirely accomplished. The waterfall model phases of software development are: requirement specification, conception, analysis, design, coding, testing & debugging, installation and finally maintenance. A good amount of time is spent in each stage of development, till all requirements are met.
Compared to this ‘set-in-stone’ approach of waterfall models, the agile models focus on ‘adaptability’ and ‘agility’ during development. As an alternative to one rigid development schedule, agile models involve multiple iterative development schedules. Each iteration goes through the entire steps of design, coding and testing. A closely-knit, cross functional and self-organizing team structure exists.
Documentation: Substantial Vs Minimal:
While emphasis is placed on documentation at every stage of software development in a waterfall model, agile methodology emphasizes increased face-to-face communication and reduced documentation.
Suitability: Predictive Vs Adaptive:
While waterfall model is suitable for development of stable programs, agile methodology is best suited for web based applications because of its iterative nature that helps in incorporating & correcting the different bugs that arise over time.
Nature: Process oriented Vs People oriented:
Waterfall Methodology is Process-oriented as it focuses on pre-planning of processes in great detail and subsequently coming up with a defined overall process to be used by whosoever uses it. Agile Methodologies on the other hand are People-oriented as they believe process definition is not an independent thing and the development of software relies heavily on the skills of the development team rather than on defined processes. Agile Methodologies use processes only to support the development team in doing their work more effectively and efficiently. Process never takes a lead in agile methodologies.
Efficiency: Low Vs High:
Efficiency is determined by the quality of software product, the lead development time and the number of bugs. Due to the adaptability of agile methods and the rigid development cycle of waterfall model, agile methods can incorporate changes and release products in lesser time.

http://www.extremeprogramming.org/.
http://www.buzzle.com/articles/waterfall-model-vs-agile.html.