In the UK, we have summer holidays also, as in the vacations that people go on in the summer or just the time off that they have in the summer, especially for students and teachers.
In the UK we don't have so many religious holidays, other than Christmas and Easter. The days off we all get at those times (and at certain other times of the year, usually Mondays) are called bank holidays, as traditionally banks and other services are closed on those days. Students and teachers also have Easter and Christmas holidays, which are usually the two weeks off around those times. As far as I know, there is no equivalent of "puente" in the UK.
When you personally do not have to work, that's called a "day off".
Hope this helps,