Because if you count the weekend to Saturday and Sunday, you can use 'Weekends'. But if you count weekend to one weekend, you can use 'Weekend'. The more accurate usage would be weekends: you refer to as all of the weekend(s), however due to normality it is widely accepted to be correct either way.

Which is correct at the weekend or in the weekend?

The usage of prepositions is idiomatic. So it varies depending on the speaker. "At the weekend", "at a weekend" and "at weekends" are used in British English; "on the weekend", "on a weekend" and "on (the) weekends" in American English.

Is during the weekend correct?

"During the weekend" is grammatically correct and has the same meaning, but it doesn't sound as natural to me. Perhaps in certain contexts "during" could sound better.