For the most part, a laundry tub is just like any other sink. The only exception to this is a double laundry tray. In the case of a double laundry tray there are two simple cross-bar strainers. They're joined by a center outlet tee that is part of the sink.
The automatic clothes washer drain may be a hose hanging into the laundry tray. To prevent having to constantly clean clogged drains in your laundry tub, you should keep some sort of strainer over these drains. This will prevent your laundry tub from becoming clogged with lint from the washing machine.
If one or both sides slow or stop draining you may be able to clear it with a plunger. However to do this you will have to stop the other side while you work. I do this with a second plunger held over the other drain.
If plunging won't work you may need to use a drain cable. The best place to work from is a clean-out under the fixture if yours is a modern house. There may also be a clean-out in the vertical vent pipe. If there is no clean-out fitting in the vent and the pipe is exposed, you may install a clean-out tee and plug.
Many houses are plumbed in such a way as to have the laundry tray drain, the automatic washer (clothes washer) drain, and the kitchen sink drain all joined into the same two inch pipe. These drains clog with a combination of fat, starch, and lint. Instead of water backing up at the kitchen sink, kitchen water backs up in the basement when the laundry sink is clogged. It may be that a floor drain is also on this same pipe, when there is, this floor drain will be the place where the slow drain will evidence itself as a pool of water appears there every time the clothes washer drains. The best solution for this problem is to have a pro run a water jet drain snake into that clogged drain. Most homes with this problem have a clogged drain there at least once a year, a good scouring though may keep you from having to clean clogged drains so often.