What Is Right? At The Weekend Or On The Weekend?

At the weekend in British English, generally. However the Americans say on the weekends when using the present simple form, to talk about their habitual routines. For example I normally go to the country on the weekend. Which is not really grammatically correct, as it should be, I normally go to the country at the weekend, or on weekends.
This is really a matter of taste, and I think people will forgive you for making this mistake, as it is not a major one.
We sometimes also say, I am going to my boyfriend for the weekend.
If you were writing a letter and wanted it to be correct I would use at the weekend, to be safe.