Yes, you can download a mobile boarding pass into your phone and use it when travelling, but not always. If you complete an online check-in, you can see that boarding passes must be printed for use
On the Ryanair's website is written, that they accept mobile boarding passes on flights from the majority of airports on their network. But some airports cannot accept mobile boarding passes yet.
Because of that, it is recommended to have also a printed version of your boarding pass.