To enter a place or situation where one is unwelcome or uninvited. He didn't want to intrude on their private conversation.
To bring in something extraneous or irrelevant. The loud music intruded into his thoughts, making it hard to concentrate.
Geology: (of magma or igneous rock) to move into pre-existing rock formations while in a molten or plastic state. Basaltic magma intruded into the sedimentary layers millions of years ago.