Guilds can only accept new applications if they are set to recruiting. If the guild is set to recruiting, you need to fill out a general application (viewable only by the applicant, guildmaster, and guild officers) and wait for them to accept it.
Guilds have the ability to invite any member to their guild, even if they aren't set to recruiting. The member must accept the invitation before the recruiting process is complete.
Note: Membership on this site has no impact on guild membership in any game. This is merely to grant members special privileges available to the guild on this site.
The main reason to list your guild is to advertise it. If someone is looking for a guild, they can easily search for guilds on their server that are recruiting and find out more about them.
While special guild support is minimal at best, there are additional features in the works for guilds, such as private discussions. Game sub-sites may provide unique features for guilds (e.g. the Guild Wars site has an option for builds where only people that belong to your guild can see them).
Some online games may have a web interface that allows you to search guilds, but they may or may not provide an easy way to get in contact with the guildmaster or other guild members. In the future, you will be able to somehow link the Gamependium guild profile to the official site's guild profile (if one exists).