The float is a part of the toilet's fill valve, and it goes up and down with the tank's water level. Different types of floats are used, depending on the fill valve's design, but the float always tells the fill valve how much water to allow into the tank. If the float is stuck in the up position, water won't sufficiently fill the tank, and you will need to adjust the float or make a replacement.
Types of Floats
Fixing the stuck float problem depends on first identifying what particular type of float your tank uses. Older style toilets used a ballcock assembly, which has a float ball attached to the end of the ballcock's long plastic or metal arm. Newer gravity toilets normally always use a fill valve that has a float collar instead of a ball. This collar is around the body of the fill valve. Remove your toilet tank lid and peer into the tank, and you can easily distinguish between a ballcock assembly and its float ball and a fill valve with a more all-vertical design.
How the Float Works
Whether the float is a ball or a doughnut-shaped collar, when the tank is filled with water, the float rests at the top of the water. Flush the toilet, and as the water level lowers as it rushes into the bowl, the float drops lower also. When the float drops down to a certain level, it triggers the fill valve into action, telling it to allow more water back into the tank. When the float reaches a certain height, signifying to the fill valve that the water level has reached a certain height, the fill valve shuts off the water.
Fixing the Problem
When the float prevents the tank from filling with water, it's because the float is not lowering far enough to activate the fill valve. If you have a float ball, verify that the ball is not rubbing against the back wall of the tank, preventing it from lowering. Gently bend the arm that connects the float to the ballcock assembly so that both the float and the arm have free movement within the tank. For a float collar, also ensure it doesn't rub against the tank's side wall, and adjust it if it does.
Replacing the Fill Valve
If the float keeps getting stuck after you adjust it, its an uncomplicated repair to simply replace the fill valve with a new one. Shut off the water to the toilet first by turning the shutoff valve behind the toilet's lower left side all the way clockwise. Flush the toilet so that any water drains out of the tank, and soak up leftover tank water with a towel. By doing this, you prevent water from spilling onto the bathroom floor when you remove the fill valve.