A coinbase transaction is a special transaction that transfer new generated coins to a miner's wallet. It's the first transaction of a block.

Whoever solves the PoW puzzle faster creates a new block and receives it reward. This reward is a newly generated bitcoin, called coinbase.
Example:
https://blockchain.info/tx/451329336cfaa7896fb4aba504bb3acd9964fc27b3fe6bbbf05615d19df313a1


A regular transaction is any other transaction, which does not involve newly generated coins.
Example:
https://blockchain.info/tx/e038d1191d09eed13ef8e4d87f95b0e3499c63aaf7bc14001e099a4e5bf50172