Synthetic oil is just oil made from 'scratch' in a way.
Instead of breaking down crude oil, synthetic oil is made from natural gas (usually) and built up into the oil.
As far as what they do, they both are pretty similar. The main difference is the oil made from crude is less perfect. And usually has cheaper additives.
Just change the oil like normal.
If you are super picky, you can change the oil a second time after a (short time) to cleanse the engine of any remaining common oil.