If the Time Machine backup disk for your Mac is full
Your first Time Machine backup includes everything on your Mac. After that, Time Machine finds and saves only new and changed items, so the backups become smaller. Also, as your backup disk fills up, Time Machine deletes older backups to make room for new ones. You may be able to use Time Machine for a long time before running out of space.
If you do run out of space, it’s best to connect a new backup disk. After you connect the new disk, open Time Machine preferences, then select it as your Time Machine backup disk.
If you connect a larger disk and wish to preserve your existing backup history, you can copy the Backups.backupdb folder from the old disk to the new one. Use Finder, and not Terminal or any other app, to copy the Backups.backupdb folder. (The folder is large, and it may take a long time to copy.)
To save space on your backup disk, you can back up fewer items. For example, you can exclude system files or the Applications folder from Time Machine backups using Time Machine preferences.
On your Mac, choose Apple menu > System Preferences, then click Time Machine.