Two main classes of mod, ignoring the mostly does nothing demo bypasses, the pointless profile mods, handful of PIRS, disc swapping mods and whatever still remains of LIVE2CON.
1) DVD mod.
2) JTAG/SMC/RGH family. I assume RSG is supposed to be reset glitch but nobody uses that term that I have seen.
There are two main classes of xbox 360 really. The original family (usually white) aka fat 360 aka phat 360 and the 360 slim (usually black, tiny eject button on the top left rather than next to the drive, big fan on the "top").
Each of those has multiple motherboard and DVD revisions. For the original form factor I would not settle for anything less than a Jasper motherboard right now (you can tell externally by looking at the power connector, preferably motherboard side as power bricks will go from old to new, also if it lacks HDMI then run away). Take any slim you like. Slims also have wireless built in if you care about such things -- I go with wired network every time really as it is faster.
DVD mods will allow you to load copied DVDs that are region compatible (region locking was optional and... odd), depending upon the game you may need or benefit from a special burner as there was a custom format released later in the 360 life that needed it but it should be easy enough to sort at least the soft launched burner there. There are some exclusives to all regions that may or may not be of interest here, and even then the region locking is still active for those old titles (with the exception of Japanese shmups most such locking was probably a means to lessen imports but that matters little as far as end results here --
http://www.abgx.net/filename/?ch=6 and
http://abgx360.cc/verified.php should give you some idea of what is out there and what goes). No modded games, no homebrew, no XBLA (you can load them as demos), no USB loading, no hard drive loading unless you have games on demand (GOD games), controller remapping without slicing up a controller, only hard drives blessed by microsoft or those few models that will take the keys and work... You can however go online (you did see bans back in the day but today it seems pretty clear).
Difficulty of 1) varies between models and in some cases firmware updates. There are also means of emulating the DVD drive and redirecting things to USB but I doubt you will find one of them (they were common enough but those that know what they are usually seek them out).
Same restrictions for the DVD emulators aka optical drive emulators or optical disc drive emulators aka ODES or ODDEs but they can have their own USB drive attached to do it. You some will have a selector you use, others made use of the picture viewer on the 360 to have a bunch of boxart there which you could view and by extension select as it knew what was read.
In any case I like to make sure you have the DVD key with the device just in case you need it. If it is reasonably high firmware now you should be OK as I doubt we will see any updates mess with things but the paranoia is there for me. There might still be banned DVD modded 360s out there if you do care about online. Do note you can download and install updates manually if you want.
2) then Ignoring king kong which most people probably converted to JTAG there is JTAG and RGH. JTAG was a simpler older mod that was blocked out in later firmwares (no way to go back really) and only applies to old style 360s. All those restrictions above about homebrew, mods, region... are gone here but you will not be able to easily get online. SMC is a term some hackers prefer for this but everybody understands JTAG.
RGH was a later mod with much the same restriction easing. It required some actual chips as opposed to the contents of any solderers junk box. Online is still hard in hacked mode (we don't generally help people do it and it is expensive) but you can have a secondary NAND aka dual NAND and go online with the clean one if you so desired, though this is an additional mod over the standard RGH install. RGH often takes longer to boot but not unreasonably so, obviously there are differences in requirements between various revisions and firmwares and when things happened but if it is prebaked then who cares. Again though for an old style/fat/phat then nothing other than Jasper motherboard here. Prebaked slim then whatever, some later chip revisions might be a bit more stable but nothing should be unusable.
A JTAG/RGH mod does not inherently allow you to play copied DVDs by the way. It will allow you to dump keys for the DVD (often the more annoying part of the DVD mod) and in some cases flash the drive too but it is not part and parcel. If you are going from scratch then you will probably be using USB and onboard hard drive loading so that likely matters little.
