When an army crosses a border into another country for battle, they are making an incursion into enemy territory. An incursion is an invasion as well as an attack.
Incursion can also be used to describe other things that rush in like an army such as an invasive species into a new region or floodwaters entering your home. When an airplane heads onto a runway it is not supposed to land on, risking airport safety, it is known as a runway incursion. And an incursion of cold air could make September feel like December.