As others had said, transaction fee is a fee to pay miners to include your transaction into the blockchain. But there are ways to send a free transaction. If you have enough bitcoin, something like 0.1 btc, you can leave them idle and accumulate maturity. Read about it on the wiki, I am still researching.
Don't attempt to send transactions for free unless you're sending many hundreds of btc, even then I would include a small fee. If you send 0.1 without a fee it will most likely not be accepted and remain in void space for a few days, untouchable and not received by the person you sent it to. Always add between 0.0001 and 0.01 depending on how urgent it is and the size of the transaction.
