However, how can Alice encrypt something to be sent over to Bob? How would Bob decrypt it? I ask because I'm curious how when I log onto my banking site, my bank sends me data such as my online statements. How does my browser decrypt that information? I don't have the private key.
This is where you're wrong; you do have a private key. As part of the handshaking process, each side generates two keys: a public key and a private key. The client sends its public key to the server, who will use it to encrypt all data sent to the client. Likewise, the server generates both keys and sends its public key to the client, which will use it to encrypt all data sent to the server.
In many scenarios, the asymmetric key algorithm is used only to exchange another key, which is for a symmetric algorithm.
