Even though Google Play is good for getting Android apps (why wouldn't it?), there are times when one can get lost in the jungle of top charts and "the best" lists and eventually miss on a good app that never made it to the Editors' Choice. That is one reason why you'd want to try other app stores. Another one is less romantic - who doesn't like free stuff, shiny things and offers? Yeap, alternative app stores surprisingly have a lot of those, which I guess comes from the fact that these stores desperately need to somehow stand out in order to not disappear into the shadow.
Not all of these stores are as safe as Google Play, though, so you have to watch out for yourself and download an antivirus beforehand. Another thing you'll have to do before being able to download anything from alternative sources is go to Settings --> Security and tick Unknown Sources. Since there is a dozen of alternative app stores, I'll only list a few that caught my attention, but you can keep digging and find something worthy.
If you are used to shopping on Amazon like I am, then it might just be logical to go app-shopping there too. Everything is neatly categorized, you can get great deals regularly, and you probably shouldn't worry about security more than when you download something on Google Play. What's different visually is larger app icons, which did distract me at the start, but I got over it pretty quickly.
There are great deals on Amazon that don't stricly offer you discounts, but may give out paid apps for free with their "free app of the day" promotions. If you are a developer, you shouldn't miss on such opportunity, why limit your exposure to a single store? Hardware manufacturers don't.
Getjar: Paid Apps for Free
Getjar is actually an app you can download on Google Play. It provides a huge library of free apps and lives by the concept "watch some ads - get some apps", basically.
You'll see some sponsored content, clearly divided as such, but that's the price of trying to get something good for free. At the moment of writing this article, I've stumbled upon latest comments by users who are no happy with the app after the last update. There are also talks of "it used to be great", so it might be losing its thunder.
F-Droid is a big repository of free and open-source apps for Android, and the community is driven and managed by a non-profit organization from England. The community is pro- open-source and free software as a whole. It tends to be more popular among developers, but if you don't care for open source, you can still enjoy a bit of free, right?