Dishwashers can leak for a variety of different reasons. However, if your dishwasher starts leaking at the same time that your sink gets clogged up, you can narrow this list down to just a few possibilities. Some of them can be remedied very easily, others with just a little effort and a few simple tools.
Dishwashers have a flexible tube coming out of the back. This tube is the drain line. It runs up to the air gap, which is usually a short metal cylinder mounted next to your sink. This air gap connects the drain line to the line leading to the garbage disposal or sink drain in such a way that water from the sink or garbage disposal can't back up into the dishwasher.
Clogged Garbage Disposal
Dishwashers usually drain into your kitchen sink or garbage disposal. If the garbage disposal is clogged then your dishwasher might not be able to drain properly. This can cause water to leak out from the air gap and to also pool in the bottom of the dishwasher. If you have an improper seal on the door, then this pooling water can also leak out onto the floor. If you have a clogged sink, a garbage disposal and any of these symptoms, then try running the garbage disposal. That may clear up both the leak and the clog. If the garbage disposal won't run, then follow the manufacturer's instructions for proper maintenance. Never stick your hands into a garbage disposal to unclog it. To prevent this problem from occurring, run your garbage disposal before you turn on your dishwasher.
If your garbage disposal was recently installed, the installer might have forgotten to remove the plug in the port for the dishwasher line. This would have the same effect as a blocked garbage disposal. To remedy this, simply take the garbage disposal back out and remove the plug and reconnect it.
Another possibility is that the house drain lines are clogged. A clog further down the line than either the sink or the dishwasher could prevent both from draining at all. This can cause water to back up in the sink and leak from the air gap or around improperly sealed dishwasher doors. This might require the services of a professional plumber, or at least a plumbing snake.