Some of the closest synonyms for appalled are dismayed and disgusted.

Dismayed implies a sense of sadness or disappointment about something that has happened. Appalled implies not sadness but extreme distaste or disgust, and the same thing is implied by disgusted.

Appalled also often implies a sense of shock or surprise.

In fact, appalled is often used alongside the word shocked, as in We were shocked and appalled. 

Shocked implies extreme surprise, but it can be positive or negative. Appalled is always negative—it’s a combination of surprise (about how bad something is) and outrage.

Appalled is a pretty strong word. Disgusted is similarly strong, though it may imply an even stronger revulsion for what has happened.

Saying that you’re appalled by someone’s behavior is about the strongest way you can say that. To further emphasize disapproval, someone might use several similar words, as in We are shocked, appalled, saddened, and disgusted by this display of violence.

Appalled doesn’t necessarily have an exact opposite. When you’re appalled about something, you’re shocked about it in a bad way (because you feel disgust or dismay about it). In this way, feeling positive about something could be considered the opposite of being appalled. Words that capture this idea include reassured, delighted, encouraged, and comforted, as in We were reassured by the results of the survey. 

The adjective appalled comes from the past tense of the verb appall, so it’s always spelled with two L’s. (In British English, appall is sometimes spelled appal, but even its past tense is spelled appalled.)