You may think that the Play Store is a fine place to get Android apps, but Verizon apparently isn't very happy with Google's dominance -- it wants carriers to have some control. Sources for The Information claim that Verizon is in early talks with both other providers and hardware makers to create a global Android store that lets developers make full use of the "specific features" of a given network. Developers would be encouraged to hop aboard by getting the freedom to advertise, and there would be dynamic app recommendations that not only suggest downloads based on where you are (like iOS), but also the time of day and friend activity. Think of it as an adaptive interface for apps you don't own yet.
The company isn't commenting on the rumor, and there's no guarantee that the discussions will bear fruit. However, the motivations behind launching such a storefront are clear. Verizon would potentially reduce Google's say over the Android app world, and could offer more apps that convince people to subscribe. It might even get a cut of each paid app, although it's not clear that Verizon is insisting on sharing revenue.
Whether or not Big Red would succeed is another matter, and history suggests that the odds aren't in its favor. Stores from platform creators, such as the Play Store and Apple's App Store, succeeded precisely because they avoided the pitfalls of the carrier portals they replaced -- they targeted broader audiences and weren't afraid to host software that competes with network services, like internet calling apps. Verizon's most recent attempt at a store shut down in 2013, in part because it was only ever offering a fraction of the content you could find elsewhere.
Also, attempts at creating app stores by committee have traditionally fallen flat. Remember the Wholesale Apps Community? It was supposed to provide a more universal app store, but a carrier-by-carrier negotiation process, reluctant phone makers and watered-down features (it was originally based on web tech) doomed it to failure just two years after it got started. Unless Verizon and crew can offer you at least as strong an app selection as what you're already getting, you might not have much incentive to change your shopping habits.
Update: Verizon didn't comment on the original story, but it now tells Recode that it has "no plans" to make this store. "Been there. Done that," spokeswoman Debra Lewis says. Don't expect a store any time soon as a result, although the source did note that these were supposed to be early discussions -- we wouldn't completely rule out a store, at least not yet.