I am hoping to expand this section greatly, but for the time being lets keep this very simple:
In people there are two (common) very clearly distinguishable types of arthritis, rheumatoid arthritis and osteoarthiritis. Hallmarks (according to a rheumatoid consultant colleague of mine) of rheumatoid are initial stiffness, whereas osteoarthritis tends to being with pain. In dogs, arthritis is most commonly osteoarthritis, very similar to that seen in humans. In all cases, there is some inflammation (so this includes swelling and stuff)... but rheumatoid is an autoimmune disease, where the body's own defence system attacks itself.
There are several treatments available, but the sad fact is none of them are terribly brilliant. Many will help a bit, prolong mobility and reduce pain, but there are no cures.
Causes: I think we would have to say rheumatoid is mostly genetic. Osteoarthritis is very clearly multifactorial. There are several genetic factors which pre-disopose to osteoarthritis, but joint injury, excessive joint usage and obesity appear to be the cause in several cases.
What can you do about it? The only thing we are reasonably confident of; is that keeping your own weight and the weight of your dog down. Keep up regular exercise, but be aware that excessive force applied to joints could result in later development of osteoarthritis.