When I use my dishwasher, after it is finished going through all of its cycles the water is staying in the bottom of the dishwasher. What is going on?

It can be one of the three things:

1. You have small food leftovers covering fine screen openings that covers pump intake on the bottom of the DW. Remove the screen, clean it and reinstall it. ( Advice: pre-wash plates before you put them inside d/w ).
2. DW is too far from the drain / sink / Garbage Disposer. As result, plumber ran too long hose between DW and highest point of the loop ( or Air Gap, if you have one.). Install inline check valve on 5/8" discharge hose to prevent discharged water from returning to your DW
3. Pump is not functioning properly. Call appliance repairman to service your DW

In addition to Milos excellent post,( I go with door number two) Your DW has a sump that never completely emptys. Regards, tom

Observations from an amateur.

1. You hope #1 is the problem... easy to fix.
2. If #2 was the problem, wouldn't it have shown up from day one of the installation?
3. Milo and SB had really good advice. I would never have thought of #2 but it makes a lot of sense.