The Sling Scope gives you a preview of where your bird is going to go before you throw it. This saves you from wasting the more important birds and allows you to prioritise crucial shots. Use it if you are running out of birds or simply have to make the next shot.
As the name suggests, the Birdquake shakes the ground all around the level, causing Pigs to move in a small space and for their structures to collapse around them. This is a good power-up to use if there are lots of pigs protected by structures, as it may well shake you a route through to them. You can also use it to dislodge pigs from tricky positions, as it does directly affect them.
This makes the bird on the slingshot much bigger and thus much stronger. If, for example, you have a stone wall you need to get through, you should consider using Super Seeds to increase your bird's potency and hopefully get through in just one shot.
The King Sling upgrades your slingshot to provide you with greater power and speed. Use it to reach targets more easily i.e. without having to take the arc of the shot into account so much, or to travel over greater distances.
The Mighty Eagle is included here as well as in the Birds section, as it functions in much the same way as a power-up. It can only be used a limited number of times, and future uses have to be purchased or earned through playing the game.