Apps purchased via the Mac App Store
When purchased via the Mac App Store, apps are installed in the Applications folder. Icons for purchased apps are automatically placed in the current user's Dock. There is also a purchases tab in the Mac App Store where you can see all of your Mac App Store downloads.
Apps purchased on another household Mac
Apps purchased on another Mac can be installed on other Macs in the household by opening the Mac App Store on each destination Mac and signing in with the same Apple ID as you used to purchase the apps originally. Selecting the Purchases view at the top of the the Mac App Store window will display all apps which have been purchased with that account. Apps not yet installed on the current Mac will show an "Install" button next to the app name and description. Clicking the Install button will install the selected app on the current Mac without charging the account again. The apps installed in this manner will be placed on the current user's Dock and installed in the Applications folder.
Sharing apps with other household Macs via File Sharing
Apps which have been shared with other Macs in your household via file sharing/copy should be manually placed in the Applications folder on the the computers they were shared or copied to. The first time you open a copied app, it will require the user to authenticate with the Apple ID which was originally used to purchase the app.