Garbage disposal not turning on with on/off switch. Water in sink draining in dishwasher.
Your Garbage Disposal and drain is backed up.  

The garbage disposal is jammed and the overload has probably tripped. 

Look into the disposal with a bright flash light and you will find it probably needs to be cleaned out. With the unit off - clean it out.  You might also want to clean the U shaped pipe - called a trap that is under your sink - it is VERY easy and if you do a you tube search How to Clean P-Trap- you can watch a video on how easy it is to do.  

Then after cleaning them you can Reset the overload button which is usually located on the side of the Garbage disposal (red button looks like a writing PENs push button) and try to run it.

All this should take you about 30 minutes including the video..