Which of the following sentence is correct?

They are different weights.

They are different weight.

'They are different weights' is correct, just as 'they are different sizes'is correct. The noun is plural, not singular, as 'they' is plural.

There is some leniency on this one I think. I prefer the plural one, but I wouldn't put a red line underneath weight either - especially if you say:

They are a different weight.

It would be a harsh grammarian to hammer you for that. But, go for the plural. That's very tidy.