Emulators

MAME (formally MESS)

I originally used MESS for TI-99/4a emulation, but MESS has since been merged into MAME.  Works pretty much the same though aside from the ROM formats and filenames having changed slightly, which is particularly annoying if you wanted to use your existing files.  Additionally the binary package doesn't include a GUI anymore so I'd suggest grabbing MAMEUI as well if your preference is to avoid the command line.

Classic99 (formally Ami99)

I haven't used this emulator too much, but it is more turn-key than MESS in that you can run a few included games right away after installing it.  Additional cartridges can be loaded in the *c.bin/*d.bin/*g.bin format (which was the format used by MESS before .rpk files).

V9T9

The pioneering TI-99/4a emulator.  After a long period of dormancy it has been reborn in Java.  I have yet to run it through its paces so I'm not sure how it differs from other emulators anymore, but I did notice that the old utilities which accompanied it are no longer included, such as the tool to convert PC99's Gram-Kracker Dumps to the format used by this and other emulators.  I'm sure all that can still be found somewhere on 99er.net.

PC99

I am just mentioning this emulator for completeness sake.  It is the only commercial emulator for the TI-99/4a and as you may have guessed not free.  I'm not sure how it stacks up to the competition and am not motivated to find out.