I wouldn't really say competition because people can sign up for both of them, but Groupon's business model is spreading.
LivingSocial works just like Groupon in that you receive daily e-mails about steep group discount deals in your area and have a limited amount of time to snatch them up. It even has a similar layout.
However, there are some advantages to being second in existence for improving that business model.
Livingsocial includes deals of the day for areas around you in your e-mail and supplies a Googlemap for the deal of the day's location(s).
It also has LivingSocial Family Edition so parents can do fun things for cheap with the whole gang.
Possibly the best perk though, is the possibility of a completely free deal.
I think this is brilliant because then OF COURSE people are going to share their deals with other people doing some advertising for the business AND LivingSocial.
There is also an Escapes LivingSocial that packages a discount weekend getaway in some pretty cool locations.
LivingSocial takes "finding the best places in your city" to the next level. They have a "356 things to do in ___" list for major cities. Users can comment on that choice supporting or refuting that claim while recommending somewhere else that is better. It's a cool forum for people to share and discover the cool things in their city.
Now if only LivingSocial and FourSquare could combine, think of that social media network...
Let me know if you sign up and get any great deals!