My house was built in 1950. The finished basement has a room that had a kitchen sink in it. At the (insistent) request of my wife, I removed the kitchen sink so we could move our washer/dryer in there. The drain for the kitchen sink is located on the basement floor. The drain is a cast iron inverted bell shape with a 1 1/2" outlet in the bottom. I moved the washer in there yesterday and installed a laundry sink for the washer to drain into. I connected the laundry sink to the drain in the floor. When the washer discharged into the laundry sink the floor outlet overflowed.

There never was a problem with overflowing when the former kitchen sink was used. I also had to drain our hot water heater recently and I used that drain without a problem. Is it possible the drain can't keep up with the volume of water the washing machine is putting out? From reading other posts on this site I get the idea I should first try snaking out the line.

Also, this same room has a pipe that comes out from the wall about 18" and is about two feet above the floor. This pipe is capped off. I opened it recently and it's dry inside. Any idea what this pipe might have been used for?

It sounds like you have a partial clog in the floor drain line. The washer discharges too much volume for the line to handle. Snake from the floor drain and see if that's better.

"Also, this same room has a pipe that comes out from the wall about 18" and is about two feet above the floor. This pipe is capped off. I opened it recently and it's dry inside. Any idea what this pipe might have been used for?"
This is a vented stubout for a kitchen sink or a lavatory. It may be possible to install a trap and move your laundry tray over to it if you wish.
Good luck, Tom

Thanks for the fast response Tom. "Speedball" is a very appropriate handle!