What's it mean when a dog wags its tail bending  down? What about straight up wagging? Tail down?  Tail straight out?

When a dog wags its tail while bending down in the front ('play bow'), it probably means 'Play with me!'  Dogs do this with other dogs as well as with their people.

Straight up wagging (or if the tail is wagging higher than the dog's topline), it usually means that the dog is happy.  For some breeds, a level tail carriage is correct, and a level wag means that the dog is happy.

Tail down, it depends.  If the dog is otherwise relaxed, it probably means nothing more.  For some breeds, a downward (or even tucked under) tail is correct in a stand (or stacked) position, and also means nothing.

However, a tail down (or tucked) may mean that the dog is frightened, uncertain, sad, or in pain if the dog also seems fearful, hesitant, dejected, or to be in physical distress.  Tail down if the dog is also growling/showing teeth, has its ears back, and/or has assumed a tense, lowered crouch position -- that's a WARNING that the dog feels threatened and will become aggressive if the situation is not immediately defused.

Tail straight out -- for some breeds (pointers and setters, for example), that's just correct tail carriage and means nothing.  Often, though, tail straight out indicates that the dog is interested in something.  If combined with a forward stance and an intense focus, it means that the dog is a little TOO interested in something or that its prey drive has been activated.  This can be a dangerous situation, too, especially if the object of the dog's interest is a baby, child, or another animal that cannot defend itself.