HTTPS stands for Hyper Text Transfer Protocol Secure - this is the secure version of HTTP, the protocol used when data is transferred between your browser and the server that hosts the website you are visiting. As the name says, in the case of HTTPS all the communication between your browser and the website you are visiting is encrypted.
This means that all data exchanged, like personal data you are filling into a webform, credit card related data, the information you are getting when opening up your Member only webpage, and so on is encrypted at one end (the browser or the server), sent across the Internet and decrypted at the other end, thus making it unusable for a potential attacker that might intercept it.
Various search engines have recently also started favoring HTTPS enabled websites when displaying websearch results. In general HTTPS enabled websites are regarded as more trustworthy for visitors.
In order to enable the secure connection, an SSL certificate is used. This certificate is purchased from a Certificate Authority by the server's administrator and is sent over the web browser of the visitor in order to initiate the so called "SSL handshake". The handshake is basically the initial exchange between the web server and the browser in which the SSL certificate that contains the public encryption key is downloaded by the browser. After the certificate is downloaded from the webserver, the encrypted communication can begin.