Good question! You might have heard of nuclear fission, this is a process that we use to generate nuclear power, and this process is just splitting of atoms. The nucleus at a centre of an atom is made of subatomic particles – protons and neutrons. To split a nucleus into separate lighter nuclei you have to fire neutrons at it. When a neutron with enough energy hits it can break the strong force that holds together the particles in the nucleus, and this causes the nucleus to separate in to two parts, making two atoms of lighter elements than the first.
Sometimes atoms split spontaneously, this is what happens in radioactive decay. The first time it was achieved intentionally by a scientist was Otto Hahn in 1938.