I am a first time new home owner and I need help figuring out what the deal is. Our home is a slab home, no basement or crawl space and we have a drain in the floor of our laundry room and it smells every time I do laundry. What is this drain for and how is it connected to everything?

When we had our washer installed, they had the drain hose just run into the utility sink and not the pipe next to the sink. The pipe is stuffed with a towel because it smells if it is not covered.

When I lift the cover of the drain there is always yucky water just sitting there.

Do I need to call a plumber to find out the smell or is this something I can handle? Any help would be appreciated.

The pipe next to the sink sounds as if there is an open connection spot with no trap. Running to the utility sink is fine if that's what you want but you need to close the open connection with a cap or plug. The floor drain should have water if its trapped. When your washer drains can you see any movement in the drain water level?

we have a drain in the floor of our laundry room and it smells every time I do laundry. What is this drain for and how is it connected to everything?
This is a floor drain. It's connected to your sewer. This floor drain has a trap and if the floor drain isn't used the water in the trap will evaporate. When there's no water in the trap the back pressure from the washer pump will force sewer gas back out of the floor drain. The solution would be to fill the floor drain trap with motor oil or anti-freeze so the trap seal would remain in place.

When we had our washer installed, they had the drain hose just run into the utility sink and not the pipe next to the sink. The pipe is stuffed with a towel because it smells if it is not covered.
The pipe next to the sink sounds like a washer stand pipe. This, also, has a trap the same as your floor drain. If you don't use the standpipe then fill that trap with motor oil as well. Do those things and I bet your smells will go bye-bye. Good luck, Tom

There is water in the floor drain, and it moves slightly when the water drains from the Utility sink. There is also a plastic piece floating in the water but it is not attached to anything, so I think that may be some sort of debris that may have fallen in there.

Take the trash out of the floor drain before it clogs it up and hook the washer standpipe and as it['s used the smell should go away. Good luck, Tom

Yep, the water moves a bit because its trapped and that is just trash like Tom says. However, I don't think we have actually answered your question regarding the smell. Come back and tell us if the smell is gone once the stand pipes is in place.