Is there any way to make an online server launch an event on an android app.

Is there any way for an online server to send information to an app. And also start an event on that app. I was thinking of something like making a listener on the android app for that website. I dont know how that would work or if that is even possible.

I was also thinking of making my original server send an email to the phone and have the phone listen in on incoming emails and perform an action when a new email comes in. Is this a good alternative? If anyone can help me with getting tutorials for how to listen for emails on an android app I would be greatful. I am also open to other alternatives.

My goal is to make my phone act as an extention of my server that just stays in one place, gets comands from another server and sends text messages from those commands.