A transaction is a transfer of value between Bitcoin wallets that gets included in the block chain. Bitcoin wallets keep a secret piece of data called a private key or seed, which is used to sign transactions, providing a mathematical proof that they have come from the owner of the wallet.
this definition is a little misleading because a transaction doesn't have to be "between wallets". it can be inside one wallet (from one address to another inside that wallet) or it can even be to the same address (like spam attacks).
also the proof is not just mathematical, it is mostly in form of an unlocking mechanism through scripts that "unlocks" the coins that are being spent. for example the HTLC contracts.