Many users, new and old alike, have been asking similar questions about ZSNES. Here are the questions and answers, condensed and simplified. For a more thorough expanation of the following, as well as many, many more questions and answers, you should read the ZSNES Documentation, or more specifically, the ZSNES Frequently Asked Questions (found in that link).
The most recent official version of ZSNES can be found here. However, ipher, one of the ZSNES developers, releases stable builds between official releases to demo new features and bug fixes. You can find these releases, called WIP (work in progress) releases, at Ipher's ZSNES WIP Site. Before reporting a bug or asking a question, be sure you are using the latest WIP release.
For anyone wanting to contribute to the ZSNES project, Aerdan has started the ZSNES Documentation Wiki project. You can find the wiki here. Please only submit useful, accurate information.
For the growing amount of novice Linux users out there, Aerdan has documented how to compile and run the ZSNES source code.
As an ordinary user:
tar jxf zsnes151.tar.bz2
cd zsnes_1_51/src
./configure --enable-release --prefix=~/bin
make
make install
~/bin/zsnes
As root (requires root access):
tar jxf zsnes151.tar.bz2
cd zsnes_1_51/src
./configure --enable-release
make
make install
zsnes
If you recieve the error "configure: error: expected an absolute directory name for --prefix: ~/bin" while compiling, you must replace ~ with the path to your home directory.