How does rotation affect a planet?
Gravity affects all objects in the universe no matter how big or small they are making any two things with mass be attracted to each other. The way this is typically described is through Newton's Laws of Gravitation where the 'Gravitational Force' is increases with mass but decreases the further you move away from it. Under extreme conditions, Newton's Law fails to match what we observe out in space but then Einstein's Laws of General Relativity comes to the rescue to explain what we see!
Now on to spinning objects - if an object is spinning, it experiences an 'outward' force - you'll have experienced this when you've gone around a roundabout in a car and been pushed outwards. This centrifugal force depends how fast you're travelling around the point at the centre of the rotation and decreases the further you go away. All planets do rotate and as such have the effects of both gravity keeping them together but this centrifugal force pulling them apart. Fortunately for us, the gravitational force is much stronger - if we do the calculations for Jupiter, the centripetal force is about 8% of that created by Gravity while for Earth it's about 0.4%. As such, if Jupiter wasn't spinning, you would have feel a force 10% bigger keeping you on the surface while for Earth, the difference would be pretty much completely unnoticeable! The condition about this however - the speed with which a planet rotates doesn't have any relation to how massive it is! It depends on how it was formed and if it has experienced anything like asteroids collisions etc.
The only thing we notice about rotating planets is that they tend to bulge out at the minute (or the technical term being oblate). This is because as they spin, they want to flatten into a disk but again, the rotation speed of the planet limits the effects of this.