The chemical elements labelled as "synthetic" are unstable, with a half-life so short (ranging from a fraction of millisecond to a few million years) relative to the age of the Earth that any atoms of that element that may have been present when the Earth formed have long since completely decayed away. Hence they are only known on Earth as the product of nuclear reactors or particle accelerators.
The first synthetic element discovered was technetium, which filled a gap in the periodic table, and the discovery that there were no stable isotopes of technetium explained its absence on Earth: its 4.2 million year half-life meant that none remained from the time of formation of the Earth.
Synthetic elements include:
Provisional names for recently observed synthetic elements: