A patch is a data file that contains instructions for changing information within a data file. In the ROM hacking community, the International Patching System, or IPS, is the most common patching format. Other patch formats exist, though they are much less common. Most ROM hacks are distributed using the IPS format; this saves space and allows for hacks to be distributed without the ROM. Distributing entire modified ROMs (as opposed to just patches) is illegal because it is copyright infringement.
More non-ROM hacking specific information can be found at Wikipedia's International Patching System article.