Some lawn mowers such as the Craftsman 37705 have a washout feature. It is essentially a garden hose fitting on top of the circular blade cover. This allows you to pump water into the blade assembly while running the engine. That action is designed to flush dirt and grass debris right out.
Place a drip pan or other container on the side of the mower with the filler. Carefully tilt the mower so oil drains out and into the pan. Discard the spent oil into a proper disposal vessel. Now slowly pour fresh oil into the mower. For the specific type of oil you need check your instruction manual. Make sure to add only as much as your particular model requires. Remember to let the mower sit undisturbed for a few minutes so the oil settles properly inside the engine.
Other items and supplies include a plastic drip tray motor oil and a container for waste oil. Often a large piece of cardboard is useful as well an old rag or cloth. Also know that this guide deals with push or self-propelled mowers not riding mowers. While some advice here applies they are another animal altogether.