There are now so many simulators available that it is not sensible to try to catalogue them.
A great deal of public domain software may be found; a good starting point is the Oakland Repository ( http://www.acs.oakland.edu/oak/SimTel/win3/neurlnet.html) which offers a range of PC products -- some of these are pretty good and others less so; this author found the nndt130 product usable. An important caveat is that while PC simulators can and do make good demos, in particular for teaching, many (most?) of them are still rather limited in performance and capability.
Those in the Unix domain will probably settle on the Stuttgart simulator (SNNS), available from ftp.informatik.uni-stuttgart.de by anonymous FTP from the directory /pub/SNNS. This product is now at version 4.0; it is mature and carries a good reputation, although has a non-negligible learning curve. This is the place to get things done, although it probably isn't the best place to get your noddy demos working.
In addition there is a variety of products around that cost real money; our bank balance prohibits us saying much about these since the only one we have bought is Neuralworks, marketed by Neuralware. This one too is mature (more than 5 years old) and has evolved into a robust and pretty product. It permits on-screen building of a variety of networks, and will dump them out as compilable code. A PC version exists.