Each banner has a different situation in which it shines. Using it effectively provides the greatest bonus, both to your team and to yourself.
The Buff Banner can, in the right situation, act as a mini-Kritzkreig for your entire team, helping them punch through an enemy front, take a capture point, or push back an attacking team entirely. It also chargest faster than any other Banner, which makes it easier to utilize in most situations, and is ideal for a defensive front since it allows your team to break apart a push.
But it won't be of much help if you don't have teammates that can deal that damage. Buff Banner is best used when you have multiple splash-damage allies on your team, such as other soldiers or demomen. Pyros can help too, as can Heavies. Basically any ally that can deal a lot of damage will help.
The Buff Banner also pairs well with an Uber medic, allowing you the benefit of both mini-crits and invulnerability at the same time.
If the Buff Banner is like a mini-Kritzkreig, the Batallion's Backup is like a mini-Uber. The reduced damage can help lead your team into a push, and the added health will help you survive that push as well. If you have a lot of allies pushing together all at once, the defense bonus this item provides can prove devestating in a push. It can even help take down that pesky Sentry Nest.
Like the Buff Banner, this item pairs well in conjunction with its opposite in Medigun-technology. A kritzkreig with the Batallion's Backup will keep the Kritz target alive while they tear through your opponents.
The Concheror, like the Batallion's Backup, pairs best with a non-Uber medic, like the Kritzkreig, since it allows the target to deal damage and heal equal ammounts from it at the same time. But it can also be beneficial if your team is lacking medics or other forms of healing, giving your team a much-needed healing boost.
In addition, both the Concheror and the Batallion's Backup have bonus effects. See Austin Mohr's answer for details.