If you have Google Apps running on your domain, now you can install third-party apps that fully integrate with Google’s apps.

Google has debuted the Google Apps Marketplace, an online store where Google Apps users can browse different cloud-based applications and add the ones they like to their suite of online tools. The apps can share data with the standard Google apps like Gmail and Docs on whatever hosting environment you’re using. Basically, you get to build your own web-based productivity suite.

The apps currently in the store (Google launched with 50) are skewed towards the business and education customers, which make up the vast majority of hosted Google Apps users right now. If you browse the store, you’ll find gobs of apps for things like project management, customer retention and administration. No games just yet, sorry.

There’s a significant amount of buzz around Manymoon, one of the handful of companies that demoed at Google’s launch event Tuesday night and currently the most-installed app in the store. It’s a team collaboration app that divvies up tasks, sets project goals and tracks the progress of team members. The browser-based image editing tools from Aviary are cool, too.

Personally, I’m most fond of eFax — certainly an app with its eye on the future of communication.

Some apps are free and some are paid, with most of the paid apps adopting a subscription model.

When you install an app, it plugs into the user account and application data you already have stored in your current suite of Google Apps. So, you’re basically adding another app to your Google stack. The new app appears alongside Gmail, Calendar and Documents in the horizontal menu at the top of the browser and there’s very little friction as far as user experience goes.

If you want to build a browser-based web app and make it available in Google’s store, the company has posted its guidelines and docs for developers.

The apps can use OpenID or OAuth for user authentication (since Google credentials act as OpenIDs), and they can access various Google Apps APIs using the Google Data protocol. So if you want to build an app that harnesses a users’ contact information, calendar events or presentations from Google Docs, the company has made it very easy to do so.

Here’s a video of the announcement from Tuesday’s Campfire One developer event, which was held at Google headquarters, featuring VP of engineering Vic Gundotra: