Time Machine is a convenient backup system and we recommend all Mac users enable it or otherwise regularly back up their data. Besides its uses, Time Machine does have a couple of quirks, and one is that after using Time Machine for a while some people may find the system has created backups for multiple systems in the "backups.backupdb" folder on the Time Machine drive.
When Time Machine runs, it will store all backup instances in a folder that is given the name of the current system, so if you connect it to a computer called "Computer1," it will create a folder "/backups.backupdb/Computer1" on the Time Machine drive in which to store the system backups. If the drive is then connected to a second computer called "Computer2," a folder for this computer will be created next to that for Computer1.
This is only done if the same Time Machine drive is being used for multiple computer systems, and is perfectly normal to see, especially if you are using network backups such as a Time Capsule. The backup folder for a given computer is not matched to it by name, but rather by other unique identifiers, such as the system's MAC address, that are stored in hidden files at the root of the Time Machine drive. Because of this, if you change the name of your system, Time Machine should also update the name of the backup drive instead of creating a new set of backups for the new computer name.
While this should be the normal behavior, some people have found more than one folder for the same computer in the backups.backupdb folder on their Time Machine drives. When this happens, the secondary folders may have a number associated with them, such as a "1" or "2" after the computer name.
This can happen because of a number of reasons. If you have had the motherboard on your computer replaced, or if you have migrated to a new system, Time Machine may recognize the hardware change as a new system and begin a new backup for it. Additionally, there may be glitches with the Time Machine drive itself that could make the hidden configuration files temporarily inaccessible and cause Time Machine to no longer recognize the system as having existing backups.
In these cases, when Time Machine runs it will back up the system again in a new folder, but since two folders in the same directory cannot have the same name, the system will append a number to it to make it unique. As a result, the secondary folder may have a couple of backup instances in it, but if at a later date Time Machine reassociates the computer with its prior backup folder, then the secondary backup folder will be orphaned.
Having multiple backup folders is not necessarily a problem unless they are taking up space; however, if you want to remove them all you really need to do is delete all but the one your computer is using. To see which one this is, just change the name of your computer in the "Sharing" system preferences and see what backup folder's name also gets changed, indicating it is the active backup directory.
Keep in mind that if you use the drive with multiple systems then there will be individual backup folders for each system on the Time Machine drive, so make sure you do not delete the backup folders that have the name of other systems you use.
If multiple backup folders keep reappearing even after you delete them, then it is likely your Time Machine drive may be experiencing problems. Some possibilities for this could be permissions errors that prevent the Time Machine configuration files from being accessed, or because of a drive or directory structure failure. To check for these errors, run a full check on the drive using Disk Utility or a third-party utility like Disk Warrior. Additionally, make sure the backup drive is formatted to "Mac OS Extended (Journaled)" and is using the GUID partition scheme (this can be seen in Disk Utility).
One solution is to just start your backups over, ensuring you format the drive properly when you do. You can either erase the current drive, or archive it and use a new one for your active backups. If you set it aside as an archived disk, you can always use Time Machine's "Browse Other Time Machine Disks" option to quickly access data on it.
In addition to starting your backups over, you can try to use a spare drive for cloning your current Time Machine disk since doing this may bypass partition or directory structure errors that could be causing problems with the drive. However, be sure you avoid doing a block-level clone since this will transfer the whole drive setup as is to the new drive.
File-level cloning can be done with Disk Utility, but more robust solutions are available in third-party utilities like Carbon Copy Cloner or SuperDuper. Before you clone, ensure the drive is partitioned to the GUID partition scheme and is formatted to "Mac OS Extended (Journaled)."