Usually, I put slices of bread in my toaster. Sometimes, I toast bagels. On my toaster, there's a little button that says "bagel". How does this setting toast the bagel differently?
Unfortunately, the function depends on the manufacturer.
While turning off a set of elements may be common, another option is that the setting may just increase the cook time. Bagels are thicker and have more mass than sliced bread, so a longer toast time is necessary to have a similar level of toasting.
In particular there are Cuisinart toasters where the bagel button only lengthens the time of the toasting and does not change the pattern.
From Cuisinart PDF manual
The bagel feature adds extra time to the toasting cycle to allow for thicker breads. You may also use the bagel button for English muffins and other thick breads.
The bagel setting deactivates or lowers the power on the outer heating elements. This has the effect of toasting the cut surface of the bagel while only warming the outer surface. Generally it's the inner elements that get up to toasting temperature.
I have a Dualit toaster and the 'bagel' setting switches off elements on specific sides of each slot, allowing you to toast only the cut side of a bagel or English muffin, which is the norm. If you wish to toast both sides, just use the normal setting.