Interpreter files are text files that are read in (interpreted) by the program specified on the first line. Gory details can be found
here in wikipedia. Shell scripts, perl scripts, awk scripts, and so forth are all examples of interpreter files. Compare that to programs that are compiled from source code files into a 'proper' executable.
Note that the file extension serves no functional purpose in UNIX - their use is a legacy of people's time on the dark side :-).
Why and how are they used has caused more debates than I care to think about. I can say "it depends". When should you write a script v. write a program - it depends. Which scripting language should you use - it depends (just not csh). The one bit of advice I have is become familiar with numerous interpreters - perl, python, bash, ksh, awk to name a few. I am sure people can name a few more.
