Choices: Why Install OR Why use Extension
An installation will allow you to store your materials on your own computer which means you have a lot more storage space.
Using both, an installation and browser extension, allows you to use any computer and still sync your work to your account.
Just using the browser extension allows you to work on any computer, but with fewer abilities then on the installed version.
How to Install
Zotero has versions for Windows, Linux, MacOS, and iOS. Some of the download options may be hidden behind a "Other options" button.
Extensions
Zotero has extensions / add-ons for several browsers. Select the browser you use most often (Chrome, Firefox, Microsoft Edge, or even, Safari), and "install" the extension.