Each sacrament (apart from ordination) is done by a Presbyter. Ordinations are done by Bishops (at least 3 must participate). Baptism in emergency situations can be done by a lay person, but it must be a Christian. I've also heard that some monks, who weren't ordained, can confess.
A slight correction to Heorhij's post: Ordination of a priest only requires one bishop, not three. Consecration of a bishop requires three bishops to participate.
Confessor-monks who are not ordained are indeed possible, and exist in reasonable numbers.