How do things stay in orbit

So gravity keeps things from floating away.. but why doesn't the object in orbit eventually spiral in toward the object it orbits and fall to the ground?

Momentum.

Each planet has momentum of movement. A planet's momentum is pointing in a straight line, tangent to its orbit. Gravity holds the planet from going off in a straight line. The two are in balance; momentum and gravity.

Another way to look at this problem is in terms of relativity. According to relativity, gravity bends space itself. Each planet moves straight ahead but its 'straight ahead' is bent into a circle because gravity is bending the space around the Sun.

Centripetal force (or maybe it's centrifugal, I get those mixed up) maintains a constant outward pull as long as the object is rotating around the planet/body (which is essentially forever, as there's virtually no friction is most orbital paths).
As long as the speed of the object and the distance it is from the planet remain properly balanced, it will stay right where it is.

Edit: gintable is right, Centrifugal is a 'virtual' force that 'pulls' the satellite outward, but is actually caused by the velocity of the satellite and the tendency of objects to want to move in a straight line. Gravity conversely acts as the centripetal force keeping the satellite from being flung into space.

It's the centrifugal force which draws the object away from the center of rotation.Also if you visualize it in general relativity's mindset,you can think as space-time as a plane and celestial bodies warp this plane around them,so smaller objects can orbit around in this warping of spacetime neither getting too close neither too far,if no extra mass or energy is added.

Contrary to what Karl says, there is no such thing as the centrifugal force.