3 main classes of hack
1) Basic USB/hard drive access stuff. Can do various things though the only ones worth really looking at are PIRS DLC. Various times the game of the year versions of games bundled DLC on the disc in the so called PIRS format. These can be installed on any 360 regardless of mod status and give you the DLC they contain. Online is possible with this. About the only other thing worth doing is game updates if one version has a glitch but not another.
Historically there was also profile sharing.Here you would get a profile that had a whole bunch of XBLA and DLC (not sure about GOD installs) associated with it, install it and the associated XBLA/DLC on your console and play. This has since been stopped, as had the modern warfare glitch and all other known ones. Not sure about Shadow Complex right now; there was an older version that you could install as a demo (oh yeah you can burn demos and install XBLA as demos if you want but it is just demos) and there was a glitch where you could get past the end of demo screen to do the whole game.
2) Drive based mods. First type is the traditional flashed DVD drive. This allows you to play region compatible games burned to dual layer DVDs. The other type is a drive emulator which will allow you to play region compatible games from a DVD emulator plugged into your 360 (most would use a USB drive), these are quite expensive.
In all cases you will have to open the 360. Some types are not terribly hard to mod and others are a complete pig, this includes DVD emulator stuff which you will still need keys for.
Banning was a thing at one point but nobody has for a while now.
3) JTAG/RGH family.
JTAG can be done with a trip to the electronics shop and some basic soldering, though it will require a 360 on a very old kernel. RGH can be done on any, though some newer types of RGH chip can do better on newer kernels and motherboard types. Soldering RGH can be slightly fiddly as well.
No online really with any of these, save for a very complex setup with RGH involving dual NAND and an even more complex, expensive and annoying setup if you stealth it.
They do however allow you to run games from the hard drive (which can be any size rather than limited to certain models/official ones, run from USB, do region free, do hacked games and cheats, do homebrew, DLC and XBLA naturally, convert hard drive installs to standalone games and quite a bit more. It won't allow you to play burned games but nobody cares as you have USB and hard drives instead.
360 homebrew is nothing special really. If you have an android box, a raspberry pi, a wii, an original xbox... you will have more to play with. A 360 with a few TB of games, XBLA and DLC installed and ready to go is a nice bit of kit if you have not played the games otherwise.
So short version, pretty much what Nyap said.