All the answers given are great answers once one has acquired some modern physics background and mathematics. For an aspiring young physicist they might be a big mouthful. So once more I will offer the simpler "everyday" interpretation.
When particles interact we talk of it as scattering of one particle with the other and have built machines that do that. Scatterings are described well with the theories developed, as in the answer by Marek, and if one means by "annihilation" that a particle disappears and becomes another particle, that is the way it can happen.
"Real" annihilation happens, is measured, experimentally when all the quantum numbers describing the two incoming interacting particles become 0 in the interaction region and the output is photons, as in the case of electron positron annihilation at low energy, and/or a bunch of particle whose quantum numbers sum up to 0 at energies that allow their creation.
So when the two incoming particles have equal and opposite quantum numbers that describe them in the Standard Model, experimentalists call the interaction annihilation if it is not an elastic scattering. Elastic scattering retains the quantum numbers of the two incoming particles, either wholly or partially. It can happen that an antiproton becomes an antineutron, retaining baryon number, for example. The processes can only be disentangled studying further with diagrams of the type shown by Marek.