The "Pause" feature is for you to use when you want to leave it and come back to it later.
As football princess just said, press (and hold down) A, then press B to put your Tama on pause.
This means that it is in "suspended animation" (!)
It won't evolve / develop - but it won't drop Happy or Hungry hearts either.
So if you can't check / play with your Tama for long periods of time (hours)... like, if you are going to be at school (for example), it is a good idea to put your Tama on pause.
But if you can keep it with you whilst you are doing other things ... like, if you have homework to do (for example), then there's no need to put it on pause; you can keep your Tama nearby and just check its stats and feed or play with it when it needs attention.
Have fun!
Best, etc.
TamaMum