Wash your hands in warm soapy water to remove as much dirt as possible. Then, soap up your nail brush and scrub under and around every nail until it's clean. The short bristles are great for really tough dirt and the longer bristles clean well under long nails and on the skin around the nails. Rinse with warm water and repeat as necessary. As a bonus, the long bristles can also be used to scrub the grout between tiles (quickly and more efficiently than using an old toothbrush).
Your brush can be washed with soap and water and can be stored on your sink, in your shower or in a drawer or cabinet. It is best to allow it to air dry before storing it in an enclosed space.