The common theme is to create something truly engaging for your market, at the right time:
Instagram was released at the perfect time when many people had iPhones with a decent camera but limited ways of sharing those photos while on the go.
SnapChat tapped into a similar (though younger) demographic that now needed a more personal space to share, as the mainstream social networks pressed forward with "no one needs privacy anymore".
I'm writing a post covering this exact topic, to be published soon.
Both platforms are exceptionally simple by design and from a UX standpoint which entices use. There are some stark differences between the two however. One is meant for global social engagement, the other is more directly personal with no public history behind the actions.
These were companies that were almost specifically designed to appeal and be sold to other services like Facebook. A need was identified (sharing pictures and videos with friends instantly) and they were simply first to apply the innovative concept and marketed it better than anyone else.
Common ideals are normally the ability to instantly share media, but usually focus on being at the right place at the right time. For example, Instagram was introduced at a point where taking pictures had reached it's peak, but being able to share them via mobile in a 'Twitter-esque' style feed was non-existent. SnapChat banked on the rise of 'FaceTime' style video chats.