How do signatures work? Just curious, if I send a transaction from one address to another, how do they verify it without knowing my private key?
Bitcoin uses the Elliptic Curve Digital Signature Algorithm (ECDSA).  Your private key is used to create the signature and your public key is used to verify the signature.  This allows anybody to verify your signature as long as they have your public key.

For more detailed information: Digital Signature Algorithm and Elliptic Curve DSA