Endocytosis is a special form of active transport used to transport material that are too large to diffuse through the membrane to be transported through membrane proteins. Endocytosis involves absorbing materials from outside the cell by engulfing them in the cell membrane. The membrane encloses the material and buds off to form a vesicle. This often fuses with a lysosome (full of digestive enzymes) to enable the digestion of the contents.
Endocytosis is often a receptor mediated process. This simply means that receptors on the cells surface will recognise particular materials and mediate their ingestion.
Exocytosis is essentially just the reverse process. Materials to be secreted are transported to the cell surface in a vesicle (a small membrane bubble). The vesicle fuses with the cell membrane and its contents are released.