Bourne Again shell. The most commonly used (and most powerful) shell on Linux. POSIX-compliant, compatible with the Bourne shell, created and distributed by the GNU project (Free Software Foundation). Offers command-line editing, history substitution, and Bourne shell compatibility.
C shell. Developed at Berkeley. Mostly compatible with the Bourne shell for interactive use, but has a very different interface for programming. Does not offer command-line editing, although it does have a sophisticated alternative called history substitution. On Linux, csh is just another name for the newer tcsh.
Korn shell. Perhaps the most popular on Unix systems generally, and the first to introduce modern shell techniques (including some borrowed from the C shell) into the Bourne shell. Compatible with the Bourne shell. Offers command-line editing.
Bourne shell. The original shell. Does not offer command-line editing.
Enhanced C shell. Offers command-line editing.
Z shell. The newest of the shells. Compatible with the Bourne shell. Offers command-line editing. Has very powerful completion features. If you do not know any shell yet, and your Linux distribution carries zsh, go with that choice from the start.