Endocytosis is a process whereby cells absorb material (molecules or other cells) from outside their cell membranes. It is used by cells (especially protists) because most substances important to them are polar and consist of big molecules, and thus can't pass through the highly hydrophobic plasma membrane. Endocytosis is the opposite of exocytosis, and always involves the formation of a vesicle from part of the cell membrane.
Endocytosis can be of three forms:
Phagocytosis is the process by which cells ingest large objects, such as prey cells or large chunks of dead organic matter, thanks to membrane folding around this objects. These are sealed off into large vacuoles. Lysosomes then merge with the vacuole, turning it into a digestive chamber. The products of the digestion are then released into the cytosol. Macrophages are cells of the immune system that specialize in the destruction of antigens (bacteria, viruses and other foreign particles) by phagocytosis.
Pinocytosis (literally, cell drinking) is the invagination of the cell membrane to form a pocket filled with extracellular fluid (and molecules within it). The pocket then pinches off to form a vesicle, and the vesicle ruptures to release its contents into the cytosol.
Receptor-mediated endocytosis is similar to pinocytosis, except it is prompted by the binding of a large extracellular molecule - such as a protein - to a receptor on the cell membrane. These receptors are often associated with the cytosolic protein clathrin, which is coating the membrane, forming a pit. When the receptors bind their target molecules, the pit deepens until a clathrin-coated vesicle is released into the cytosol.