Anybody can burn a disc and play XBLA demos (works on a stock machine) but that is besides the point.
DVD games can be done by flashing the DVD drive or emulating the DVD drive. These days for most current games (and most future ones) you will need a specialist DVD burner as well (it is known as XDG3 if you fancy reading up).
DVD flashing is covered rather nicely in the tutorials
http://jungleflasher.net/downloads.html
Do note that you can dodge some of the harder key extraction methods by getting the CPU key and NAND image to decrypt it all from that (you will probably end up heading that way so before you splash out consider that) and there are standalone flashing tools available.
Various drive emulators exist now- some have been reviewed recently around here as well.
You may encounter some drive hacking chips- if not for a 360 slim them ignore them. If you do hack the dash it will not allow you to burn games and play them (the DVD drive is the gatekeeper).
They can also be played on hacked dash consoles (methods to this are JTAG and glitch) either by extracting them and running them with a specialist program or converting the iso or installed version to GOD format (a format you can download games from but was turned to use for this). JTAG is only possible on machines not updated since 11th August 2009 (and made before mid June of that year) and as such are rare, glitch is fairly new but can be done on just about every model out there (I am not sure about the newest of the new). Give or take they have the same abilities but glitch requires custom hardware (not expensive but not as cheap as JTAG) and is not quite as reliable (although had I not said that it you might not have ever known) where JTAG can be done with very basic components (as in those available from your half decently stocked electronics shop).
JTAG
https://docs.google.com/View?id=dnfmv5h_30dw33vpf4
Glitch is pretty similar (certainly the later stage software side of things is save where JTAG uses Fbbuild this uses GGbuild) but installation varies slightly depending on the chip you buy and you might also have to update the chip used to launch the glitch attack.
XBLA and DLC can be done with two main methods- the first is the hacked dash method detailed above. Here you install them to the hard drive or a USB drive set up for use on a 360 (the up to 16 gigs of your USB drive as memory card stuff)
The second and one I do not favour (not to mention I am not sure if it is still available) is shared profile. Here people would get a profile, buy a slew of XBLA and DLC, strip the identity info from it and upload the relevant XBLA, DLC and profile for others to run (this required access to live so banned consoles did not work).
A few glitches and such have happened over the years (stuff like the modern warfare 2 glitch) but for the most part we ignore them.
Some DLC comes in a form called PIRS- usually from game of the year versions (stuff like the fallout 3 one, borderlands double pack and a few others) that you can inject these onto your hard drive or your USB drive.
Short version- you want to hack the DVD drive and do either JTAG (unlikely you have a viable machine) or glitch to get the best of all worlds here.
