This is where Booker and Elizabeth come in. You can think of Elizabeth as the x-factor in the conflict; both sides want her because of her abilities. She can manipulate rips found throughout Columbia. These rips are tears in the fabric of time and from them she can bring various objects into existence. The tears can help or harm Booker and Elizabeth in the conflicts they face as they traverse through Columbia.
As important as Elizabeth is, Booker has his own tools of the trade and with the help of American ingenuity he can utilize bottles of Vigors and Nostrums. A spinoff of Bioshock's Plasmids and Tonics, Vigors bestow Booker with a number of abilities like a shock ability that electrocutes and stuns enemies, while Nostrums function as permanent boosts to Bookers abilities like increasing his health. That is a familiar mix of powers and guns, but this is the new generation of Bioshock and there is more still.
In a floating city you must have some way of getting around. Initially there were skylines built around the city that were said to transport cargo, but people have learned to get around on them with a skyhook. With said skyhook Booker can traverse around a level either to get where he needs to go or he can use them tactically in combat.
Imagine a classic FPS encounter — you are presented with enemies in front of you with various objects in-between to give you cover. Then it's a basic point and shoot until you drop all the enemies and move on. Keep imagining, now you have these skylines that you can use to get above, below or to the side of your enemies. Instead of hiding behind cover and methodically taking out baddies, you can choose to hop on the skyline, move to a platform where the enemies are out in the open, and take them all out at once with a handy grenade. This mechanic allows for you to be smart and allows level designers to build interesting maps by constructing encounters to be more dynamic. In one example from the E3 2011 demo, Booker has to use the skylines to board an airship to destroy it from the inside and the scene finishes with him dive bombing out praying to catch a skyline before hitting the platform.