The absorption of substances into a cell, by forming pockets in the cell membrane and engulfing molecules or larger objects from the extracellular fluid that adjoins it.
The pocket becomes a kind of bubble called an endosome within the cell. Its inner and outer membranes are reversed compared to those of the main body of the cell.
There are three forms of endocytosis, called pinocytosis, phagocytosis, and receptor-mediated endocytosis.
Pinocytosis means cell drinking, and is the normal continuous process that all cells undertake, of sucking in small droplets of liquid. This gives them a sample of the ambient chemicals and enables them to react according to what is there.
Phagocytosis means cell eating and is when a much larger bubble is formed (called a vacuole or phagosome). This is done only by specialized cells (phagocytes) when there is something of interest in the extracellular fluid (ECF), such as a bacterium that needs to be destroyed.
The phagosome so formed transports its captive inside the cell to another organelle, called a lysosome. The two organelles fuse their membranes and the contents of the phagosome are destroyed or absorbed by the contents of the lysosome.
The third kind of endocytosis is mediated by receptors on the surface of the cell. These have strong affinities to particular substances the cell needs, such as iron or cholesterol. Endocytosis is induced when the molecule is detected in the ECF. This can be thousands of times more efficient than pinocytosis.
Source and further reading: I learnt about this mainly from this webpage, which is clear and not too difficult:
http://people.ne.mediaone.net/jkimball/BiologyPages/E/Endocytosis.html