Okay, so far, the best game play video is here: http://www.youtube.com/watch?v=rKtYqWBR7h0
This is another video, but is not game-play: http://www.youtube.com/watch?v=jzVjggarRns
This is a video from EVE Fanfest 2012 showing it off: http://www.youtube.com/watch?v=dDW-HrzBo7s
Basically, this a 'capture the flag' type of game. Basically how this fits into EVE is that a Corporation (guild) in EVE Online decides that it wants to claim sovereignty, or control, of a solar system. To do this, they must complete a series of steps: They have to set up a player owned station and, once DUST 514 comes out, must claim control of all the planets inside that system.
To do this, the EVE Corp creates a contract for DUST 514 Merc Corps to assault a planet. Once the contract is accepted, it initiates a series of capture the flag type of missions, ultimately granting control of the space elevator. A bit more about the ultimate goals and the game features here: http://www.youtube.com/watch?v=-Sdbq_KeOlA
Now, in these videos, you'll see that vehicles are in the game as well. How they've chosen to do vehicles is pretty unique. Each vehicle gets inserted via dropship individually. The number of vehicles you have is limited based on how much money you have and how many 'points' you have to allocate to them. How these 'war points' work, I have no idea, they haven't put that info out yet.
Now, the unique thing about these vehicles is that they are fully customizable, just like ships in EVE: http://www.blogcdn.com/massively.joystiq.com/media/2012/03/dust514vehicle.jpg and http://img4.mmo.mmo4arab.com/news/2011/11/29/dust514/dust514_vehicle1.jpg. This means you can fit your vehicles to accomplish a vastly wide range of roles, all decided and engineered by you.
So far, not much more than that has been released by CCP or Sony, but from interviews and videos, I can also say that the Mobile Mission Command (MMC) ship can be purchased and customized just like the vehicles can. This means, inside of your ship, you can do all kinds of things. In EVE, there are leadership skills and modules. Each one gives a bonus, some to the amount of armor/shields you have, some to your targeting speed, damage, or firing speed.
Now, how this ties into EVE is DUST 514 players can request Orbital Bombardments. These come to EVE players as missions. The EVE players then have to get their Battleships into orbit, on the actual correct side of the planet, target the location, and initiate the bombardments. This will require a massive amount of coordination, which CCP plans to happen through their built in voice and text chats.
I can also imagine that they will have uses for other ships, such as Logistics and Command Ships. Logistics ships can remote repair Armor or Shields. I can see them doing this from orbit onto the MMC ships, which have their *own* battles. Additionally, the Command Ships give bonuses to Armor/Shields and Damager/Rate of Fire, and all kinds of other things.