In the latest version (Loki 0.4 Stable), I can't open .deb files with AppCenter, like Ubuntu.
Use Eddy (the recommended, graphical, elementary way)
sudo gdebi package.deb
sudo apt install gdebi
Then open the package using
apt (the proper cli way)
sudo apt install ./package.deb
dpkg (the way that doesn't resolve dependencies)
sudo dpkg -i package.deb sudo apt install -f
Note that attempting to install a .deb that is inside an .iso can cause errors, so you may want to copy it to your downloads folder.
One easy way to install Debian files easily is to install Eddy. Run this command to install it
sudo apt install eddy
As Seth noted out in the comments, the previous does not work in Elementary OS 5.0 Juno. Try this instead:
sudo apt install com.github.donadigo.eddy
To install a Debian package with Eddy first you open it and it will show this page:
If you downloaded the Debian package into your Downloads folder, you can click the "Load from Download" option. It will then show the screen below. Then you can click install on whatever package you need to.
Or, you can select the "Open" button. It will show the file manager and you can choose what file you want to install.
You can use the options already provided. Or you can install Gnome Software.
sudo apt install gnome-software
Gnome Software is the new shop for
Ubuntu 16.04. With it you can install the
.deb package, and the same list these softwares installed, options that the AppCenter doesn't have.
The Gnome Software does not disable AppCenter.