As an adverb meaning "very" or "extremely":
* "The movie was awfully long and boring."
* "I'm awfully sorry I missed your birthday party."
* "She felt awfully tired after a long day at work."
As an adverb meaning "badly" or "terribly":
* "He was awfully hurt when she broke up with him."
* "The weather was awfully cold this morning."
* "The accident was awfully tragic."
As an intensifier, similar to "very" or "really":
* "That's awfully kind of you."
* "He's awfully handsome."
* "It's awfully nice to see you."
The meaning of "awfully" can be subtle and depend on context. It can also be quite informal, so be mindful of the setting and audience when using it.