There are three main classes of hack
Minor software fiddling.
Things like installing PIRS DLC (DLC that comes with some game of the year games and can be installed by anybody, quite legitimately, on anything). You can do this with a USB drive. If you get into modding saves and maybe dashes you might run into issues, every other hack that fell into this category (profile sharing and such like) has since been squashed by Microsoft.
DVD mod
Varies a tiny bit depending upon what dash you are on and varies quite a bit depending upon what model of 360 you have.
Also falling into this are ODDE type mods where the DVD drive is intercepted by a not so cheap device and redirected to USB.
With these online is available, you can be banned by at this point we have not seen bans in years but play it like it might happen and if you buy a year of live and it happens tomorrow you will get little sympathy from anybody.
Limitations as far as new games. Some future games might see something odd happen which sees you have to update your DVD mod though this is considered unlikely at this point (it can and did happen at several points but we are drawing towards the end of the 360's lifetime).
You can download them yourself, you will need a special drive (see ihas and related drives as well as BurnerMAX) to burn many newer games though. You are restricted to games made for the region of the xbox or otherwsie region unlocked for your region. Europe is actually pretty good these days as long as you are not a fan of playing every Japanese shmup that comes along and are not a fan of American sports though even there you can get a lot done.
JTAG/RGH/RJTAG mod
This is the good stuff -- it allows USB loading, homebrew (there is not a lot), hard drive loading, region free, modded games, XBLA, DLC and a bit more besides. You are looking at finding someone that knows how to solder and spending a bit on parts for any of this save perhaps JTAG though your machine is probably long past JTAG capable (August 2009 was the update that killed JTAG).
It does not inherently allow copied DVDs but with the other options most people do not care.
It comes at the cost of online, there are some very annoying mods for the RGH stuff that allow you to maintain two kernels and have one for online and one for hacks but most around here would not suggest doing this as it is not easy to maintain a working/safe setup here.