You don't necessarily need a database. You need a common network protocol between two applications.
All network communication is done via sockets. You need a library that allows you send data over sockets. For example, here's an Android guide that is about sockets.
A socket binds to a specific port of a computer, essentially making it a "server". Much like how web servers all expose port 80, and communicate over a protocol called HTTP. Which is important because it is up to you to decide what protocol your applications communicate between each other, because the socket just sends bytes - it doesn't care what you send or how, as long as it travels to a port on a particular server. It also won't parse the data for you, that's up to your application to handle. For example, how would your desktop app know the Android device sent it a text message, or some image to be displayed, or an address to show a map?
All in all, your reason for wanting a desktop application rather than a web application is not entirely clear. Parsing only the body of HTTP payloads from different HTTP paths that are mapped to different methods (which is typically referred to as a REST API) is much simpler than building your own protocol. You might as well build a desktop GUI over top of a web server.
Making the desktop app send updates back to your mobile application is basically impossible using a bi-directional socket architecture. Your Android should not be running an open server socket continuously just for your application, mostly because battery drain, but because its network address is subject to change frequently, and you therefore additionally need a registration server from which your device would reconnect to. Such a service exists as Firebase Cloud Messaging, which is a rebranding of the GCM technology made by Google, and it can be used to send push notifications to devices, but only with small data payloads.
See here about what activities occur on an Android device for notifications. How does push notification technology work on Android?
Back to the question about databases. Suggesting one to use is too broad. And you only need one of those if you want to store and/or query or join datasets. The same computer running the desktop app can install and run whatever flavor of database you prefer, whether it's a relational database or noSQL database, entirely up to you. The only realtime databases I know of are RethinkDB and Firebase.
You could also just hold a SQLite file which is as good as a small scale database (even the SQLite documentation recommends it for low traffic web sites).
