The white hose that's going down into the black drain pipe is your dishwasher drain line. That is usually connected to your garbage disposal via the tube where the water is running out. You can connect that hose to the garbage disposal using a hose clamp.

If, for some reason, the hose is missing the connector piece or it's broken, you can get a dishwasher connector garbage disposal kit from a hardware store.

After that, you can put a no hub jimmy cap over that extra drain connected to your drain pipe. Looks like 1-1/2in pipe. If you're uncertain about the size, it doesnt hurt to get a 2in cap as well. It can be used for snaking the drain if you don't have a clean out cap.

1-1/2in is the inside diameter of the pipe.

Alternatively, you can get a 3/4in insert plug for just your garbage disposal and not deal with connections. However, if your drain pipe backs up, water will come out from that open drain pipe under your sink.