To investigate is to look into something systematically. If you don't understand how tornadoes work, for example, investigate the forces of wind and find out.
Investigate still acts a lot like the word it comes from, investigationem meaning "a searching into," which came from the Latin for "to track a footprint." It's exactly what you do when you investigate. You can investigate in the name of science, which is to look into something systematically, like when you research how tornadoes work. If you investigate a person, you find out everything you can — and you might need to hire a private eye for help.