Here is a very simplified explanation:
Your web browser downloads the web server's certificate, which contains the public key of the web server. This certificate is signed with the private key of a trusted certificate authority.
Your web browser comes installed with the public keys of all of the major certificate authorities. It uses this public key to verify that the web server's certificate was indeed signed by the trusted certificate authority.
The certificate contains the domain name and/or ip address of the web server. Your web browser confirms with the certificate authority that the address listed in the certificate is the one to which it has an open connection.
Your web browser generates a shared symmetric key which will be used to encrypt the HTTP traffic on this connection; this is much more efficient than using public/private key encryption for everything. Your browser encrypts the symmetric key with the public key of the web server then sends it back, thus ensuring that only the web server can decrypt it, since only the web server has its private key.
Note that the certificate authority (CA) is essential to preventing man-in-the-middle attacks. However, even an unsigned certificate will prevent someone from passively listening in on your encrypted traffic, since they have no way to gain access to your shared symmetric key.
It's also worth noting that in addition to purchasing a certificate (as mentioned above), you can also create your own for free; this is referred to as a "self-signed certificate". The difference between a self-signed certificate and one that's purchased is simple: the purchased one has been signed by a Certificate Authority that your browser already knows about. In other words, your browser can easily validate the authenticity of a purchased certificate.
Unfortunately this has led to a common misconception that self-signed certificates are inherently less secure than those sold by commercial CA's like GoDaddy and Verisign, and that you have to live with browser warnings/exceptions if you use them; this is incorrect.
If you securely distribute a self-signed certificate (or CA cert, as bobince suggested) and install it in the browsers that will use your site, it's just as secure as one that's purchased and is not vulnerable to man-in-the-middle attacks and cert forgery. Obviously this means that it's only feasible if only a few people need secure access to your site (e.g., internal apps, personal blogs, etc.).