When you do something with purpose, you do it with determination. When your activities have a purpose, you have an aim or intention in mind.
This noun also has a third meaning: "function, role, or use." The purpose of a fork, for instance, is to spear and scoop food from your plate, not to poke your little brother at the dinner table. Purpose can also be used as a verb meaning "to reach a decision" or "to intend," though these uses of the word are quite a bit less frequent nowadays.