Error: backup disk is full - all nn possible backups were removed, but space is still needed. This is pretty clear.  Time Machine deleted as many old backups as it could (and they're all listed in the first failed backup's messages). See item #C4.

Bulk setting Spotlight attributes failed or several Waiting for index to be ready.  There may be a problem with your TM drive, or difficulty communicating with it.  See item #D2.

Error: (-nn) SrcErr:YES Copying {a file path} to {"null" or another file path}

or Indexing a file failed. Returned -12 for: {a file path}, {another file path} These may indicate a problem with the first file referenced.  See item #C3).

Found partially deleted backup;  trying again to delete.  The named backup is incomplete -- most likely, Time Machine tried to delete it previously, but it's damaged.  If you see this repeatedly, try deleting it via the Finder.  If it still won't delete, try Repairing your backups, per item #A5.

Ownership is disabled on the backup destination volume.  Enabling.  This is normal on the first backup to a new or freshly-formatted volume, as ownership will be enforced.

Backup content size: xxx MB/GB excluded items size: yyy MB/GB for volume zzz.  These sizes are rough estimates (and sometimes quite far off).  On initial backup only.

Using file event preflight for <name>.  (Mountain Lion 10.8.x and later.)  Time Machine was able to use the File System Event Store to figure out what changed and needs to be backed-up.  See the blue box in How Time Machine works its Magic for an explanation. 

. . . xxx MB/GB requested (including padding), yyy MB/GB available.  The amount requested is more, at least 20% more (sometimes much more), than the estimated size of the backup, as Time Machine needs extra workspace temporarily on the TM drive/partition.

No pre-backup thinning needed.  Time Machine has room for the new backup.

Starting pre-backup thinning usually followed by:

No expired backups exist - deleting oldest backups to make room and a list of deleted backups.  These are weekly backups that TM must delete to make room for the new backup.

Inherited root volume <name>, UUID: <long character string>.   (Lion and later.)  Time Machine has "reconnected" to your backups after your Mac's logic board was replaced, or you did a full system restore or transfer from another Mac via Setup Assistant or Migration Assistant.

First backup after disk inheritance . . .  (Lion 10.7.x and later.)  Time Machine has "reconnected" to the backups of a previous disk (or after a disk has been erased), after a full system restore or transfer from another Mac via Setup Assistant or Migration Assistant.  This will be followed by a Deep Event Scan, per the blue box below.

No post-backup thinning needed.  There are no expired backups.

Starting post-backup thinning, followed by a list of deleted backups.  These backups are either hourly backups over 24 hours old, or daily backups over a month old.  See Time Machine FAQ #12 for details.

Copying Lion Recovery set followed on Lion by Backed up Lion Recovery to /Volumes/<name>/Backups.backupdb.  Effective with Lion 10.7.2, the first backup to a directly-connected external HD with the GUID partition map scheme copies the Recovery HD to the TM drive.  See the green box in Using the Recovery HD for details.

Event store UUIDs don't match  naming your internal HD or any other drive/partition being backed-up.  Time Machine can't be sure the OSX internal log of file changes that it normally uses is correct. 

This happens in the following circumstances:

1. •The first backup of a disk, or to a new location
   

2. •After booting from your install disc, Recovery HD, or a different OSX volume (usually)
   

3. •A full restore
   

4. •Certain hardware repairs (e.g., new internal disk or logic board)
   

5. •Removal of large exclusions
   

6. •A very large volume of changes (especially an OSX update)
   

7. •An improper shutdown of your Mac, or improper disconnection or power loss on the Time Machine drive
   

8. •If it's an external HD, an improper shutdown or disconnection of the drive
   

9. •A failed or cancelled backup
   

10. •Many days without a backup
    

It usually causes a lengthy deep scan/traversal (below), so if you see it frequently, you need to figure out why it's happening.

Deep event scan followed by Finished scan (Lion 10.7.x and later)

or . . . node requires deep traversal.  (Snow Leopard 10.6.x. and earlier):

Usually you'll see this after one of the UUID messages above.  Instead of the log of file changes Time Machine normally uses, it must examine every file and folder on the named drive, partition, or folder, and compare it to the last backup, to figure out what's changed and needs to be backed-up. 

Or, it may reference your iPhoto Library, mentioning a "missed reservation," meaning the iPhoto app was open on the previous backup, so Time Machine couldn’t back it up then, and will try to "catch up" now.  If it can't, it will try again on the next backup.

Note: if files are changed by a different operating system (such as OSX Tiger or earlier, Linux, or MacDrive on Windows), no entries are made in the log, so Time Machine must do a full scan to find the changes.

Obviously, this may be a lengthy procedure;  and especially lengthy if you're doing wireless backups.  As this is part of the "Preparing" phase ("Calculating changes" in Snow Leopard), you may not see any more messages for quite a while.  Try not to interrupt the backup, as this must be done again (and again) until a backup is completed successfully.  On Snow Leopard only, effective with 10.6.3, you'll also see a running count of items that Time Machine is "scanning," under the progress bar on the Time Machine Preferences panel and the menu from the Time Machine icon in the menubar.

Waiting for index to be ready

A few of these may not be a problem; your backups need to be indexed for Time Machine to work properly.  But several may indicate trouble.  See below.