Phagocytosis is a form of endocytosis. In the process of phagocytosis the cell changes shape by sending out projections which are called pseudpodia (false feet). 

The phagocytic cell such as a macrophage may be attracted to a particle like a bacteria or virus by chemical attractant. 

This process is called chemotaxis (movement toward a source of chemical attractant). 

The phagocytic cell sends out membrane projections that make contact with some particle.

Some sort of receptor ligand interaction occurs between the phagocytic cell surface and the particle that will be ingested.

The pseudopodia then surround the particle and when the plasma membrane of the projection meet membrane fusion occurs. 

This results in the formation of an intracellular vesicle.