To aspirate is to exhale or breathe air out as you speak. Many languages include word sounds that require the speaker to aspirate.
Many speakers aspirate as they pronounce certain sounds, and in linguistics or speech therapy, it's a common term. In medicine, to aspirate is something very different — it means to suck or breathe in, often in a dangerous way. Someone who drowns, for example, aspirates water as they struggle. The Latin root is aspirare, "to breathe upon, blow upon, or breathe."