A biological invasion occurs when a species rapidly colonizes a new geographical area. The new area is often very far from the regions considered to be part of the species' native range.
Well-known examples include the invasion of the southern states of the U.S. by fire ants (originally South American), zebra mussels (originally eastern European) in the Great Lakes, the dispersal of cane toads (originally South American) in Australia, and grey squirrels (originally North American) in England. I've written about invasive species before, focusing on the example of fire ants.
Many invasions are not instantly successful, and don't really get going until quite a long time after a species is first introduced to a new geographical area. This is called a lag. This phenomenon may seem mysterious. Some alien species seem to cling by their fingernails at low numbers for years, before suddenly exploding into invasiveness.