Decay means that particles change into other particles. This happens when an original particle no longer exists and gives up its energy to form other particles (known as daughters). An example of this is when the top quark decays or becomes two particles, a bottom quark and a W boson.
Annihilation can occur if a particle and an antiparticle come together. If they are annihilated, they both disappear and their energy makes new particles. An example of this is when an electron and its antiparticle (positron) come together. They can create a photon.