Synthetic elements are made by adding protons or neutrons to existing elements. A beam of subatomic particles is fired at an atom of an existing radioactive element. When this atom captures an extra proton, or captures an extra neutron and decays, a new element is formed.
The most common process used to create synthetic elements is adding additional neutrons. It is easier for the atomic nucleus to accept the neutrally-charged neutron than a particle with an electrical charge. After it joins with the nucleus, the extra neutron undergoes beta decay and becomes a proton. This extra proton changes the atom from a heavier isotope of the base element to an entirely new element.
When protons are added directly to the atom by firing a proton beam, a new element is formed without the intermediate step of beta decay. This is a more difficult process than adding neutrons, as the positively-charged nucleus and protons repel one another.
Uranium is the most common base element used to create synthetic elements. It is also used to create elements that exist in very small natural quantities, such as plutonium. While these elements do exist in nature, they are produced in larger quantities using this same process.