What's the difference between sending and receiving address? I mean, if for mistake I asked to send a payment to my sending address, have I lost my bitcoin? TX
Addresses can both be used both for sending and receiving transactions. However, some online wallet services share addresses between different users, so you want to be careful if you are using one of those. Use a "receive payment function" or "my address", or whatever it might be called, to get the address that can receive payments for you.
If you are using the standard client, all your addresses are fully under your control and are safe to receive transactions.
There is no "sending" address in the Bitcoin world. If you send to an address you have seen a transaction of yours from on a blockchain explorer, it's possibly yours, possibly not if you were using an online wallet like Mt. Gox's. You should avoid doing that at all costs, as the result is at best unpredictable.
If you are using software similar to BitCoin Core-Qt, sending addresses are simply addresses that you likely do not have the associated private key. For instance, I might send BTC to my friend and he has provided me an associated address to do such. They may have created a "Receiving Address" on their BitCoin Core-Qt and provided it to me say through an email. I would put that address in my "Sending Address" list with a note showing its purpose, e.g. "To Dillon for the Superbowl Beer". These notes do not show in the blockchain, only in your wallet.