What server-side language should I use for mobile apps? (Apps that are like Snapchat, which are not like Facebook...

Mobile backends that only deal with I/O (input/output) and don't do much CPU-bound activities (calculations, parsing, etc) are often written in JavaScript these days, for node.js. I believe LinkedIn is an example of such server-side architecture. Yahoo and PayPal are also moving a great deal of their architecture to node.js.

Snapchat itself appears to be written in Java, another popular choice, due to Java's popularity in general and ability to scale.

Theoretically, you could write the backend in anything, even PHP would do for a small-scale app, although I wouldn't recommend it.

You could also use a BaaS like Parse or Backendless, or an open-source self-hosted alternative like BaaSBox. In that case the backend is more about modelling the date than development.

The thing is, there is no golden rule. Usually people use whatever they're most familiar with and then, if necessary, move on to a different architecture (like Twitter which started as a Ruby on Rails app and then moved its backend to Scala).

Snapchat runs on Google App Engine Backend.