If you are going to only use the database from your desktop, you don't need PostgreSQL installed on your laptop so you could just download the Upgrade version.
Configuring your desktop to allow anybody to connect to PostgreSQL will take a bit more work; you'll need to make sure your router forwards port 5432 to the desktop. You'll need to know the external IP address of your desktop (go here and it will tell you). You also need to configure your pg_hba.conf with a slightly different line. You'll want to use 0.0.0.0/0 and make sure you use md5 there, not trust. If you used trust there anybody anywhere on the internet could log into your PostgreSQL database without a password.
Then you connect to your desktop from your laptop using the external IP address and it should work.