Most interactive fiction (AKA text adventures) games require an interpreter. Using an interpreter typically means that the games are smaller and can be run on a wide variety of systems. Indeed, the old Infocom text adventure games ran on Commodore 64s, Apple IIs, IBM PCs and many other systems. Thanks to modern interpreters you can run those same games on just about any modern computer while getting higher quality fonts.
I recommend getting a Z-Code interpreter immediately; it's the most common format. Track down other interpreters are you run into a need for them.
If you're looking for some games, well, there are a bunch available for free on the IF Archive. The archive itself is a bit tricky to navigate, so I recommend Baf's Guide to the IF Archive. Trying checking out the 5 star rated games. For something shorter (about 2 hours per game), I recommend checking out the first few places in the IF Comp. I cannot recommend the fairly long horror title Anchorhead enough. You can check out my reviews of IF games for more suggestions.
Z-Code games use the same system as the old Infocom games. Indeed, most Z-Code interpreters can run most or all of the Infocom games. Developers still use this format as it's portable and highly efficient. (An old 4 megabyte Palm device can easily run these games.) This is perhaps the most common format.
For Mac OS X, I highly recommend Zoom for Mac. It features "iFiction", an interface for managing your games similar iTunes.
For Linux (and other Unixes) I recommend Zoom for Unix. You might need to fiddle with it to get anti-aliased fonts.
For Palm devices, there are two good options. CliFrotz supports graphics quite well., but wants to use little teeny fonts. It does run games directly off a memory card. I have a slight preference for Frobnitz, which doesn't support graphics and requires games to be run through a preprocessor, but is simple and lets me have big fonts.
For other systems, I cannot offer any advice. This is a fairly complete list of interpreters.
TADS (The Adventure Development System) is a relatively new format, but very popular for authors. There is really only one interpreter. There are two major versions, 2 and 3. If you're using Windows, download the TADS 2 and 3 Player Kit you should be good to go. For other systems, Check here for TADS 2 support.