The verb "to implement" means, according to my dictionary,
put (a decision, plan, agreement, etc.) into effect
and it's from Latin implēre (to fulfill), from which also "employ" derived through French.
So "an implementation of TeX" is something that makes TeX (and connected software) usable on a particular machine.
"TeX, the program" is publicly available free of charge and everybody is entitled to use the source for producing derived works. A restriction is that the final product cannot be called "TeX" if it doesn't pass the so-called "trip test".
A program in source code does nothing, what's needed is to adapt it to a processor architecture and to an operating system, so that it can be compiled and made into an executable program. Let's go simple, first, and consider only TeX in the original Knuth form.
One needs to prepare a "change file" where some machine dependent parts must be filled in (I/O and access to time data, for instance). Then TeX can be compiled (often after first translating the Pascal source into C) and run.
However, TeX (the executable) by itself is almost useless. One has to teach it how to access vital parts, such as the font metric data and input files. This is usually done by linking it to a library called
kpathsea (by Karl Berry) during the preparation of the change file or in the compilation process. This library facilitates file system access by TeX and also running auxiliary programs when needed (for instance calling METAFONT when font metric data is not available). It's not the only way: the KerTeX project uses a different library.
Having TeX in good shape is still not sufficient. One needs a DVI previewer, a way to produce the needed bitmaps for the fonts, a printer driver, METAFONT. On modern systems, the DVI previewer and the printer driver would be linked to a library for using Type1 fonts.
And that's not the whole picture: nowadays a TeX system should provide pdfTeX, XeTeX and LuaTeX, whose compilation and packing are not easy tasks, considering also what's needed in order to make the complete bundle usable on a machine.
Some "standard packagings" emerged already in the first years of TeX. In the olden days one could get a tape containing prepackaged distributions of TeX for some mainstream architectures: Unix (in some of its variants), Vax/VMS, IBM mainframes. When PC's appeared, some distributions became available on diskettes, free of charge or commercial: emTeX for DOS or OS/2, OzTeX for Mac OS, AtariTeX, AmigaTeX were on the "free" or "shareware" side; PCTeX, Textures and others on the commercial side.
Nowadays the most used implementations of TeX and friends (that is a bundle of connected software for using TeX or variants thereof) are, on the "free" side, TeX Live (Unix based architectures, but also Windows) and MiKTeX (Windows only; a GNU/Linux version had been announced, but never really implemented, as far as I know).