Electron-positron annihilation is the process that occurs when an electron (which is matter) and a positron (which is antimatter) collide. If they are both relatively at rest (not a likely scenario, practically), they destroy each other upon contact, and produce two gamma ray photons of 511 keV each which are emitted in opposite directions. If they are moving at different relative velocities, the energies of the photons emitted will be higher, in accordance with the conservation of 4-momentum (see below for important details). At higher relative velocities, other particle-antiparticle pairs can also be produced, since there is enough kinetic energy in the relative velocities to provide for the masses of those particles. At much higher relative velocities, there may be enough energy to produce a single neutral Z boson.
Photon pair production
An important point is that there must be at least two photons produced if photons are produced, since photons are massless particles. This means that their 4-momenta have a norm of zero. However, if there are two photons moving in different directions, their 4-momenta are added before taking the norm, so one ends up with a positive net 4-momentum. This 4-momentum must be the same as the 4-momentum of the colliding electron and positron.
The opposite process is known as pair production.
See also