Infer and Imply

To infer something is to guess at a meaning without a lot of data.  If John is frowning, you might infer that he is unhappy.

To imply something is to suggest that it is true without explicitly saying so.  Have a look at the following example.

Susan said, "There was a cookie here a moment ago.  Then John walked by, and now the cookie is gone."

Susan has not accused John of taking the cookie, but she has implied that he took it.  John might infer that she thinks he took the cookie.