Is standing water at bottom of dishwasher normal?
Standing water in the bottom of the dish washer after a cleaning cycle is complete is NOT normal. After the wash/rinse cycle and the pump out your dishwasher not have water in it. 

Standing water could be due to number of problems. I'll list out some of the things to check on.

1) There is a possibility that the filter screen in the bottom of your washer is mostly plugged up and not allowing the water to drain down into the pump. Clean it and try another wash.

2) Many dishwashers are installed with an air gap assembly at the kitchen counter level. These are designed to keep the sink drain from siphoning or draining back down into the dishwasher. Sometimes things from the dishwasher can get up into the air-gap and plug it and thus impeded the drainage of the dishwasher.

3) Your washer could be old and tired from washing countless loads of dishes. The pump unit in the bottom that drains the water out can sometimes fail and need replacement.

4) With some older style dishwashers that have the rotary timer knob it is possible that some electrical contacts in the timer mechanism get burned and make certain parts of the wash cycle not work correctly. This could be causing the drain pump to not be running when it is supposed to. Replacement of the timer would be the fix for this. 

Note that dirty food bearing water left standing the bottom of the dishwasher could be a spawning ground for all kinds of nasty and unhealthy things. Can also leave something to be desired about the quality of the wash job it will do on your next load of dishes. So for the health of you and your family it is a good thing you are looking into this problem.