Marble is a very porous stone, so time is of the essence when it comes to removing stains. Immediately blot all spills, avoid rubbing as it may cause spreading of the stain. The longer a stain sits, the more likely it is to soak in and will be nearly impossible to remove. Here is one removal technique that will work for several common stains.
Try washing the marble with dish liquid. If the stain is on the surface, this may remove it. If it does not, continue with the steps below.
Another option is to use white vinegar. Soak a cloth with the vinegar and lay the cloth over the stain for several hours. Remove the cloth, clean off the area with water, and repeat the process if needed.
If washing the stone with dish liquid does not work, make a poultice with the dish liquid. Add a squirt of dish liquid and a small amount of water to 1/2 cup of an absorbent powder, such as flour or cornstarch, to make a thick paste. Spread the paste over the area, then cover the area with plastic wrap. Leave the poultice on the stain for several hours or overnight, then take off the plastic wrap and allow the paste to dry. Gently scrape off the paste once it is dry and repeat the process if needed.