In this answer about mountaineering on a glacier, @StrongBad mentioned 4th and 5th class belays. I have never heard of belays being classified into such classes. In what context are these classes used and how are the different classes defined?
The belay class in the linked answer is in reference to the typical methods one would use in grade 4 (or 5) Yosemite Decimal System.
Typically, in a rock climbing context you typically see grades like 5.6, 5.10a. The 5 indicates that the route is what we commonly call "rock climbing", but grades 1-4 also exist. These are for hiking (class 1), scrambling (2 and 3) and easy climbing, but way easier than the UIAA grade IV.
If you decide that you need to do a class 4 climb protected by a rope, you commonly would rope up as a party and simul-climb while the first person places intermediate protection (such as nuts, friends, or, in the context of the linked question, ice screws). In contrast, the typical method to do class 5 climbs is to have a belayer that belays a leader (i.e. typical rock climbing).
Of course, this explanation only takes typical cases into account. You might need to belay children or handicapped people on class 4 terrain, or you might decide to simul-climb a YDS 5.5 (UIAA IV) climb under certain circumstances (if you know what you are doing!).
Note: As the comments point out, the terms might not be commonly used. For a 4th class running belay, the commonly used term is running belay, not 4th class belay.
A better description for 4th class belay would be simul-climbing or running belays. That is when you are roped together with intermediate protection. I described such a scenario here. Its typically done on easier climbs when the risk of falling is much lower.
5th class is actual rock climbing with belaying from anchors or the ground. One would belay from the anchor while the other lead climbs up and set a anchor to belay the belayer up to. This can also be called pitching it out.