There are a wide range of mint errors, some of them common and due to inadvertent sloppiness at the mint, and others that are specific and repeated for certain coin series. There is no way to globally give you a run-down of mint errors as most of them are specific to each coin and year.
We list the most common errors for most coins. The best thing you could do to get started is to peruse our info pages and read up on whichever coins you may posess:
Gold Coins
Silver Coins
Wheat Pennies