Older mid-sized homes often times have linoleum installed as the flooring in a laundry room. These floors can often times wear out especially with heavy washers and dryers and especially if there were any previous water leaks. It is a good idea to replace the flooring in the laundry room with a simple ceramic tile or waterproof vinyl. This material can be relatively inexpensive, but is very durable with the heavy appliances and is perfect in areas that can get wet.
Here is a basic list of items to consider for your laundry room floor.
The first part of the process is disconnecting the washer and the dryer. Make sure you shut off the hot and cold water supply valves to the washer, and disconnect the connection to the dryer vent duct in the back. You probably need the help of another person to move these appliances to another room.
Then you should demo the existing flooring. If you have baseboards in the room, be careful removing them. Then comes the prep work if you are going to put in ceramic tile or vinyl floors. After the prep work, you need to establish your pattern of laying down the tile or vinyl.
Move the washer and dryer back into the room and re-connect the water supply lines and drain lines for the washer and the vent duct for the dryer.
Here is a video from Lowes using an Armstrong vinyl product.