After flushing, water continues running from the tank into the bowl.
When this happens and the tank does not refill correctly, the trouble is in the rubber flush ball valve. In most circumstances the flush ball would be worn so that it doesn't seat the right way against the flush valve seat. Because it is made of rubber, it eventually turns spongy or ragged and replacement becomes requisite.
To replace this ball, shut off the water supply first to the toilet tank. If this is inconvenient, similar effect can be attained by tying the float ball arm as high as itwould go with a piece of string tied to an object overhead. This would keep the inlet valve closed so that it won't refill when the tank is emptied. Empty the tank by flushing, next, hold the ball stem in one hand and unscrew the ball from its lower end with your hand. Before replacing with a new one, clean away the exposed metal valve seat by rubbing gently with a piece of fine steel wool. When the threaded stem rod on which the ball fits is rusted or bent, it might be a good idea also to put in a new ball stem at the same time.
Turn on the water supply to check the entire flushing operation. If the new ball doesn't seat correctly after the tank has emptied, it may not be centered properly over its opening at the bottom. This can be made by a bent ball stem rod or by a stem guide which is a bit out of alignment. If the stem is bent, it's best to replace it. If the stem guide is out of alignment, it is easily adjusted. Undo the set screw which holds it in place, then jiggle the guide back and forth until it makes the ball to drop directly onto the center of the seat. Tighten it permanently in this position.
Tank fills properly after flushing, but water continues to pour in until it runs out through the overflow pipe
The usual trouble here is in the float mechanism or in the inlet valve which it controls. Try lifting up on the float arm a bit using one finger. If this ceases the hissing noise and shuts off the flow
Unscrew the float ball from the end of its arm and shake it to see whether there's any water on the inside. If there is, the ball is already defective and must be replaced. If the ball is not leaky, the trouble must be because of improper float arm adjustment. To correct this, bend the wire arm a bit downward in the middle so that the ball is about 1/2-inch lower inside the tank. Turn the water on and try flushing again. This time the water should shut off when its level reaches to within about 1 inch from the top of the overflow pipe. If it still goes on rising, try bending the float arm a little lower.
If pulling up on the float arm fails to shut off the flow of water
The trouble is then in the ballcock itself. The washer on the inside could be worn or the plunger might be clogged with rust and dirt. To disassemble most models, pull out the two thumbscrews or pins. These release the entire float arm mechanism, as well as the attached linkage at the top of the ballcock. The stem and plunger can then be hoisted out to expose a leather washer at the bottom of the plunger. This washer could be a force fit in its recess, or it might be held in place with a single screw in the center. Most models also have an additional washer or O-ring which fits into a groove around the body of the plunger. To be safe, this ring-shaped washer must be replaced at the same time.
Badly corroded ballcock mechanism parts that break while being handled
Your best bet is to replace the entire unit. Ballcocks cost only a few dollars, and a new one would assure trouble-free operation for many years to come. Hardware stores now sell better modern versions which are totally silent in operation and which come complete with all needed connecting fittings, and also complete installation instructions.