All stars in our galaxy are in stable elliptical orbits around the galactic centre. But they are not all moving in the same direction with the same speed... meaning there is a random maxwellian distribution of velocities among the stars.
What this effectively means is (like the animation that Crazy Buddy posted) although there is a net effective attraction between the stars and the centre of the galaxy, whenever two stars get close to each other, they exert a sort of "gravitational drag" force, which acts like a frictional force and ends in a net slowing down of the interacting bodies.
When the star then slows down, it doesn't have the speed to maintain it's current orbit, so it will move into a nearer orbit closer to the centre of the galaxy. This is the general mechanism of how stars in a galaxy are "collapsing" into the centre.
There's no need to worry about falling into any (hypothesised) super massive black holes though. Any of these processes take a really long time, and by the time we are close enough to the centre of the galaxy to worry about it, our sun will be well into it's red giant stage and the earth would already have been consumed by it. :)
P.S - To clarify my first paragraph, assuming there are no other stars orbiting the centre of the galaxy, then we will never spiral into it. That effect is only caused by the presence of other stars in the galaxy.
EDIT: @JohnRennie pointed out that because of conservation of momentum, the lighter stars will tend to gain energy and the heavier ones loose energy in dynamical friction interactions. This will tend to push the heavier stars closer to the centre and the lighter stars further out. His answer to this same question points this out.