A Mod-Chip is quite simply a chip that Mods the Xbox bios or fully replaces the original bios with a Modded bios. Without these pupies we wouldn't be playing back-ups of our games or be able to run homebrew software.
There are many types of Mod Chips out there. Older mods had so many wires to solder that you would think an octapus just took over your Xbox. were as the newer mods are requiring less and less wires and some even no wires or solder. Also newer modchips tend to be full bios replacements instead of just patching the original Bios.
Every game that you purchase for your Xbox is signed with code written by MS. When you put the disc in your xbox, the bios see's that it is signed by M$ it allows the game to load and play. When you Back-up a game or even create a game homebrew style the M$ Signature is gone and the M$ Bios will reject the disc. This of course brings the need for a modified Bios.
If you cannot solder I would recommend purchasing a no-solder mod-chip such as the Matrix/Xodus. Currently the Matrix and PcBioxx/OpenXbox are not shipped with a Bios at all and you will need to obtain a Bios such as Evo-X 2.5 (Do not mistake the Bios with the dashboard software!) and program it into the mod-chip. Other ModChips such as the Xecutor come with a Modified Bios pre Installed on it. Now you ask what is a Bios???? Find the post with in the FAQ's