Updated by Andrew Hall
The public key (address) to your wallet is not something that needs to be kept safe as it is just a string of characters representing the location of your wallet. People can use this address to send you funds. In fact, if you were to make a transaction on the blockchain, anyone would be able to view your address on a website such as etherscan.io.
Example public key: 0xC2D7CF95645D33006175B78989035C7c9061d3F9
Every public key has an associated private key. The private key controls funds associated with its corresponding public key address. Do not share your provide keys with anyone you do not trust or authorize to send funds. Also, if your private key is lost, you will no longer be able to access funds associated with that key.
Example private key: 3a1076bf45ab87712ad64ccb3b10217737f7faacbf2872e88fdd9a537d8fe266
Whenever you conduct a transaction on the blockchain, you are signing that transaction with your private key. Signing the transaction proves it came from you.
Further reading from around the web: