What is this part of the sink called and how can I make it stop leaking excess water when I run the dishwasher?

Looks like an air admittance valve --- replace it.

Likely the dishwasher empties into the side of the garbage disposal and where the disposal sits it may be broken seal

Often times the garbage disposal will sling gunk into the tube at the connection to the disposal/drain. Disconnect the tube from disposal and clean it out with a swab. The same thing happens with Reverse Osmosis filter systems which have a wash water disposal tube connected to the drain. The connection will fill full of gunk.

See Mr. Smartypants, pretty good explanation. Look up anti-siphon because water in the sink would just go back to the clean dishes by gravity. Pump hose is at the bottom of the dishwasher, sink drain is two feet higher. Plastic tubing is clogged from this device to disposal or drain causing it to relieve pressure at the valve.

Call a plumber. They are not cheap but worth the money.

That's an anti-siphon device. When your dishwasher finishes its cycle it pumps all the water out, through a tube, into your garbage disposal. There's a tube that comes from the dishwasher to this gizmo, then from the gizmo down to the garbage disposal. The little tower is to keep water from siphoning back from the drain to the dishwasher, which it would otherwise do if the garbage disposal is full of dirty water.

The problem you have is that the tube is clogged, either fully or partly, between the anti-siphon and the garbage disposal. This causes the dishwasher drain water to back up and come out of the anti-siphon tower. You fix this by taking that tube off, from under the sink to the side of the garbage disposal (have a bucket under it). Then blowing out the tube, hopefully somewhere outside. Then reinstalling it.

It's the air gap. Check to make sure that the hose coming out of it is not kinked or clogged, and check that the garbage disposal, or wherever it drains to, is not clogged.