If you've ever experienced a broken toilet tank, whether it is a break in the internal componentry or crack in the tank, you'll suddenly find yourself in a predicament. Ideally, you'll want to replace the tank (or broken parts) as quickly as possible. However, if you need to special order a tank or wait for a slow superintendent to get to fixing it, it may take a few days. If you only have one bathroom, this is a clear problem.
Get a bucket. You're essentially looking to simulate the action of the toilet tank releasing gallons of water into the bowl, so you'll want a bucket that holds a few gallons. A mop bucket would be perfect. If you don't have a bucket, use a garbage pail or can.
Store the bucket next to the toilet for handy use.
When you desire to flush the toilet fill the bucket, using the bath faucet for quickest fill.
Dump the bucket into the toilet bowl. You will need to dump at a slow but steady pace. Too slow and the water in the bowl will not flush completely and you'll have to repeat. Too fast and you risk splashing yourself--obviously an undesirable outcome.
Repeat as necessary to clear the bowl.
Use as often as you like. This method will keep your toilet bowl functioning and clear while you wait for the proper parts to fix the broken toilet tank. It can save you money that you would otherwise be forced to spend purchasing a complete new toilet (bowl and tank). It will also save you from having to run to the nearest public restroom every time you need to use the bathroom.