Is there a wallet, which can send and receive bitcoins without the need of having downloaded the complete blockchain?
Yes, these wallets are called SPV (Simplified Payment Verification) and the most popular one,
at least from this community, is electrum
[1]. Generally, you don't have to download the entire blockchain if you want to just use a wallet, but I should mention the downsides of following that way;
- Less privacy. The node you'll connect to update your balance and broadcast transactions will know your IP and which addresses you own. (You can face the IP exposal with Tor)
- Less security. By verifying the transactions you don't have to trust anyone's words that what he's telling you is true. Your computer's verification speaks by itself.
[1]
https://electrum.org/#home