Headhunter will see Spartans collecting a skull for each kill they get, a skull that then needs to be deposited into a moving skull receptacle. You're able carry multiple skulls at once, and to fight and use abilities while carrying said skulls -- but if you get capped before arriving at the aforementioned skull depository, say goodbye to that big win.
Stockpile has two teams fighting over four neutral flags, only able to "hold" said flags at the end of each of the round's five minutes. Multiplayer designer Lars Bakken notes, "If you see the other team holding three flags and there's 30 seconds left, you better start sending a couple dudes over to at least get them out of the capture point when the time runs out."
Generator Defense (or "Network Test 1 Beta") introduces Elites into the mix, with Spartans defending a trio of generators while the enormous mandible-laden creatures storm up a hill to destroy them. "This [mode] takes place exclusively on Overlook in the Beta. Defenders have the superior position at the top of a hill, with Attackers trying to claw their way up it," Bakken explains. This mode takes places only on the map "Overlook" in the beta and features smaller teams of three Spartans versus three Elites.
Finally, Invasion is only scantly detailed, though the folks from Bungie allow that it'll take place on Boneyard and will feature lots of vehicles (also: it's huge). The dev also mentions that all of these modes won't initially be available in the beta, which will have "five different playlists active at different points." If the past few days have been any indication, we'll assuredly have more details for you before the beta's May 3 opening day.