Could someone tell me how repair is supposed to work in New Vegas? More specifically, through your pip-boy? What skill do you need to be able to repair through your pip boy, do you require an item?

Just press R when you have more than one of an item, or a repair kit.

So you need to either have one of the same weapon or a repair kit, then where does the skill factor in? Can the repair kit be used with unique weapons?

Skill determines by how much the item is repaired.

In Fallout 3, skill also determined the maximum condition to which you could repair an item, but that doesn't seem to be true in NV; e.g. at 50 repair I am able to repair my weapons to 100%.

The repair kit can be used on unique weapons. Unique weapons can also be repaired using the standard model for their type, e.g. the Ratslayer uses standard Varmint rifles to repair.