Your smartphone may very well be the hub of your digital life, but there are still times we all need to sit down at a computer to get some work done. At times like that, you don't want to constantly pick up the phone to check notifications, retrieve files, and send messages. There are ways to avoid that by installing a few apps that let you control and manage your Android device from a computer. Your options vary a bit based on how you balance control and convenience.
One of the core features of Android is the notification framework. Google has added some awesome new features in the last few revisions, but there is one that doesn't get a lot of praise -- the notification listener. This feature was introduced in Android 4.3, so you'll have to be on that version or higher to take advantage of most of these apps and services.
The notification listener provides a secure way for an app to mirror notifications in both directions. That is, duplicate the notification from the system UI, and also tell the system when a notification has been dismissed so you don't see it in the regular notification shade as well. This framework can be used to duplicate notifications anywhere -- even on a desktop.
There are a few apps that can be used to monitor all your Android notifications on a computer, but the most popular might be Pushbullet. This app is good for more than just keeping track of your notifications. It can send text, links, addresses, and files between your devices as well. To set up the notification service, just make sure you have the Pushbullet extension running in Chrome or Firefox (yes, that's required) and enable Pushbullet on the phone or tablet you want to monitor.