If you found this searching you most likely found out about arbitration mailboxes much the way most people will. By either finding out they accidentally deleted them, or by finding out that you need to move, disable or remove them in order to delete a database, or uninstall Exchange 2010, or remove the mailbox role.
"Arbitration mailboxes are used for managing approval workflow. For example, an arbitration mailbox is used for handling moderated recipients and distribution group membership approval."
This is part of the Moderated Transport features that are new in Exchange 2010.
A lot more information about using arbitration mailboxes can be found here: Understanding Moderated Transport
In short, arbitration mailboxes are where messages awaiting moderation are stored, as well as information about moderator decisions are kept.
Now back to getting over the two most common immediate needs for arbitration mailboxes.
I deleted my arbitration accounts from my AD
This isn't really all that bad. I did it the first time I installed Ex2010 and had a panic moment before I found the fixes. Pretty simply, you need to rerun the AD preparation steps from the 2010 media.
Only /prepareAD is required to recreate these accounts, but I left the other steps in here as well just for documentations sake.
I am trying to remove Exchange 2010, or a database, or the Mailbox role and am being told there are arbitration mailboxes preventing me from continuing
This is also not too bad. When you try to remove the first DB in Exchange 2010, there are a few arbitration mailboxes that will prevent database deletion. You have the choice of moving, removing, or mail-disable these mailboxes. Since you cannot see these in the Exchange Management Console, you need to launch Exchange Management Shell (EMS)
This will list the arbitration mailboxes. To narrow it down to a specific database, you can edit this to:
Get-Mailbox -Arbitration -Database DB1
If you are used to PowerShell cmdlets in Exchange 2007, one big change to recall here is that specifying servername\databasename won't work anymore. This is one of the reasons why the database names need to be unique to the organization - so you don't have to specify servers anymore!
Once you have your "get" command returning the correct list of mailboxes, it's time to move, disable or remove them. Disabling the last arbitration mailbox is not allowed, so I recommend moving them as the first preference here.
Get-Mailbox -Arbitration -Database db1 | New-MoveRequest -TargetDatabase db2
Get-Mailbox -Arbitration -Database db1 | Disable-Mailbox -Arbitration
Get-Mailbox -Arbitration -Database db1 | Remove-Mailbox -Arbitration -RemoveLastArbitrationMailboxAllowed
If there is enough interest a little later, I may do a write up on using the arbitration mailboxes, but at this point there is still a lot of other Exchange 2010 things to learn and figure out!