IF THIS THEN THAT is a free online service that allows you to connect many different online services together. The idea is simple, if a certain thing happens on one web application, then do something on another. For example, if someone tags me in a photo on Facebook, then save a copy of that picture to my Google Drive account.
The IFTTT community can create if this then that connections, called Recipes, between lots of different applications, called Channels, including Facebook, Twitter, Dropbox, Evernote, Gmail and others.
IFTTT also works with devices like the Belkin Wemo, UP wristband by Jawbone, Withings, and the Philips Hue bulb.
Get a more comprehensive overview of IFTTT on their about page.
The power of this service comes from mixing and matching applications and actions. If you use a few different web services online, you might find a Recipe to automate something you're already doing manually. Browse from an ever-growing collection of IFTTT Recipes.
Here are just a few to give you an idea of what this web service can do.