Social Anxiety is a pathological condition pertaining to consistent sets of behaviour and responses - fear, anxiety and discomfort - deep inset into an individuals psyche, due to negative past experiences, and the triggering of which is caused by social situations and interactions. Since it has become ingrained, it is usually extremely hard to get rid of, and doesn't simply go away without proper help to correct it.
Shyness is insecurity and general uncertainty surrounding interaction with people, similar to social anxiety. It is usually down to lack of confidence and a sensitivity to rejection, and is quite often simply due to personality traits (genes or observation of parents behaviour). Most shy children loose it because of repeated interactions - so the brain becomes accustomed - and some keep the traits through to adulthood (usually milder), but others yet keep it and they sometimes become avoidant, then the condition is developed as the brain becomes wired to automatically trigger those feelings of anxiety due to the number of negative associations with social situations; then SA is "coded" into the brain. One difference is that SA is akin to/a form of phobia and is irrational, whereas shyness is quite natural and a rational construct (especially with children).
And being an introvert has nothing to do with either, no matter which definition you take, although that's not what most people would have you believe. Introversion produces an outward behaviour that is superficially similar to the above, but the actual causation is very different. A popular definition is an individual who is most predominantly concerned what is going on inside their own head rather than with external reality. That usually makes them quieter and less bothered about interacting with others, yes, but of course that doesn't make them the same thing. A shy or socially anxious person is usually quiet out of fear and apprehension, rather than purely about interest with themselves more than their environment. Of course, they aren't mutually exclusive, so one can be shy/have SA and be an introvert or an extrovert, and there are many introverts that aren't shy etc. Another definition is where one gets energy from; introverts from themselves, whereas interaction/their environment saps it. Like the previous definition, it often creates similar outward behaviour, but that doesn't make them shy or socially anxious.