4 Blocks.
So, the "attacker" would need to solve 5 blocks in a row faster than the rest of the network in order to double-spend the funds. It would cost a lot to attempt it and the chances of success would be very slim. I think after 6 confirmations you can consider the transaction safe from a double-spend.
1 confirmation is about 10 minutes so 6 confirmations take about an hour. Your wallet or blockchain.info will tell you how many confirmations have happened.
If you buy from someone on localbitcoins.com and do a face to face transfer of funds for Bitcions then you do not have to wait at all. The Bitcoins are transferred instantly from the sellers account on localbitcoins.com to your account on localbitcoins.com
See my response here for a guide to localbitcoins.com:
https://bitcointalksearch.org/topic/m.3087780