The correct meaning of aweful or awefully is: "inspiring awe; solemnly impressive"; or meaning "Extremely bad or unpleasant; terrible."
These words are often used intensively in the sense of "notable, notably, very, exceedingly"; but such usage is slang, and should not be followed in careful speech or writing. It is informal to write, "He was awfully sick" or "The boat was awfully fast."
