A protein is a molecule comprised of long polymeric chains of amino acids that fold in complex ways in three dimensional space. The side chains of these amino acid subunits have different chemical properties and they can interact with each other to aid in protein folding or they can interact with other chemicals to give binding interactions. Proteins are often embedded on the surface of cell membranes. When a signal molecule (often called a ligand) binds to a specific protein on the cell membrane, the change in conformation of the protein sends a signal through the cell membrane and into the interior of the cell. This can cause a cascade of chemical events and interactions within the cell to change the activity or function of the cell. This process is called signal transduction.
Another way that ligand binding can alter a cell's activity if by allowing certain molecules to enter or exit the cell. If a channel protein spans the cell membrane, then the binding of a ligand molecule can cause the channel to allow a particular chemical species to enter the cell through the membrane. You can think of it as the ligand causing a small hole to open in the cell membrane that is specific to a particular chemical.