Convection is a process where heated fluids expand, making the particles spread out, and making the fluid less dense, causing less dense fluids to rise above cooler more dense fluids, and these cooler fluids then heat up and rise upwards, creating a cyclic motion of particles, known as a convection current.
Conduction in a metal is a process where a metal consists of positive ions and free electrons, as temperature increases, the ions gain kinetic energy and vibrate, this energy is then passed on via collisions to free electrons which can move throughout the whole metal, these electrons transfer kinetic energy (again via collisions) to ions further through the metal, increasing their temperature.