Mandolin Truss Rods
A truss rod is placed in the neck, beneath the fingerboard and helps strength the neck and resist the pull of the strings.
Not all mandolins are equipped with adjustable truss rods. Those without them usually have a simple hardwood stint inlayed in the neck to offer some rigidity.

Multiple pieces of laminated wood were used to strengthen (and decorate)
this mandolin neck which has no adjustable truss rod.
As the tension from the strings strive to pull the neck upward, into a bowed shape, the truss rod is used to exert tension in the opposite direction.
A bowed neck, with too much relief not only raises the string height (action) it can also change the intonation.
Checking The Necks Relief
A straight edge can be placed down the center of the neck, while tuned to pitch, to determine the amount of relief.
You can also use the string as a straightedge. With the mandolin tuned to pitch fret the G string at the first fret and with your opposing hand fret the last fret on the neck (not the one over the body). You can now judge (or measure) the gap between the bottom of the string and the fret halfway between the two.
You may also place a capo on the first fret to free one hand to take measurements with a feeler gauge.
Adjusting Truss Rods
Most mandolin truss rods are accessible beneath a small cover found on the peghead. Tightening the truss rod nut counteracts the pull of the strings, straightening the neck.
All adjustments to a truss rod should be done in small increments and re-checked. 1/4 of a turn can have quite an effect on some necks.
When a truss rod is loosened it is allowing the string tension to pull the neck upward (into a bow).
Correct Relief
Most mandolins will play cleanly with a near dead flat neck. Necks that change frequently or are subject to humidity and temperature changes may serve the player better if a small amount of relief is present to counteract those changes.
The amount a neck will change is as unique to that instrument as the wood used to make it. Some mandolins many never budge and others may require frequent adjustments until the neck settles in.
In general, very light gauge strings may require additional relief due to their wider vibration pattern which may permit them to hit the frets and buzz when strummed hard.
Problems
Maxed Out Truss Rod
My truss rod is as tight as it will go and the neck still has way too much relief in it.
A maxed out truss rod is quite a dilemma. When these can be repaired, fingerboard planeing and refretting with large tang fret wire is used to straighten the neck. If, however, the neck shows itself to be weak and spongy, satisfactory results may not be obtainable using that method.
This is a situation that requires physical examination to determine the best course of action.
All adjustments to truss rods should be done in small increments. On some instruments 1/4 of a turn could have quite an effect on the relief. If you see that no changes are occurring when tightening the truss rod and it becomes difficult to turn it is time to stop! Some have continued tightening an unresponsive rod only to end in with a pop and a tear.
Broken Truss Rods
A broken truss rod can occur due to over tightening or, on older inexpensive instruments ...cheap metal that's destined for failure. Because truss rod adjustments are inexpensive I would recommend asking for help if there is any doubt about it's function.
A broken truss rod is usually an expensive repair and only considered on value instruments as it requires fingerboard removal (replacement in some cases) and refretting.