This depends on which browser you are using.
For Firefox there is an option called 'Firefox Sync' which will back up and allow you to synchronise your bookmarks through several devices. This is found by clicking the three horizontal bars at the top right and choosing Sign in to sync. You will need a Firefox account but these are free and last forever.
Google Chrome has book mark sync to turn this on. There is an option that looks like a person in a circle to the left of the three vertical dots at the top right. If you click this symbol there is an option to start sync. This will prompt you to create a Gmail account if you do not have one but if you do you can just log in to your Gmail account.
Each of these browser and other browser have a more in depth guide on how to do this in their own section.