To implement something is to do it as opposed to planning it or agreeing to do it.
E.g. UN resolution 242 was swiftly passed, however the implementation is proving difficult.
The root word is implement, so implementation is where the implement is applied to the surface.
In computer programming project methodologies, implementation refers to the phase of actual coding as divorced from requirements gathering and design which come before it; and integration, testing, rollout and maintenance which come after it.
E.g We have implemented according to specification.
However in reality is is hard to draw such sharp distinctions.
Writing code can be a form of design, of expressing a design and testing its doablity, so implementation is favoured by some programmers as a method of design. Any design that has not been put to the test of coding is not proven to be sound - issues often come up during coding that were not forseen, so design without coding is problematic.
Also, code that is not integrated and tested is not proven to be suitably robust, and thus some integration and testing will occur during implementation. And some more coding (i.e. implementation) will occur as a result of testing.
One the most interesting new methodologies, Extreme programming denies that implementation can or should be partitioned off as a separate activity from testing and design. Open Source development on some of the largest and most successful projects such as Mozilla and Linux proceeds without a clearly delineated "implementation" phase that is without design or integration.
Excessively linear software development models are the sign of excessively hidebound organisations. Avoid them if you can. Worst are those companies in which design is completely divorced from implementation, designers never stoop to implement and implementers are never allowed to design.
When there is a standard, an implementation can refer to a particular instance of that standard, which is functionally the same (or mostly the same) as others, but which of course will have it's own strengths, quirks and weaknesses.
e.g. BSD's implementation of TCP/IP is great or Without the fancy graphics, Diablo is just a reimplementation of nethack.
In some object-oriented programming languages there is a separation between interface and implementation parts of classes. Even when there is not, a particular section of code may be referred to as "interface code" or "implementation code".