Generally speaking, in modern Christianity, Protestant and some independent Catholic churches allow for ordained clergy to marry after ordination.

In what religion are priests not allowed to marry?

Throughout the Catholic Church, East as well as West, a priest may not marry. In the Eastern Catholic Churches, a married priest is one who married before being ordained. The Catholic Church considers the law of clerical celibacy to be not a doctrine, but a discipline.

When did it become forbidden for priests to marry?

It was not until ecumenical meetings of the Catholic Church at the First and Second Lateran councils in 1123 and 1139 that priests were explicitly forbidden from marrying.