Thumb-under is the move where the thumb passes beneath the other fingers to carry on up the keyboard. It exists for an arithmetic reason: a hand has five fingers and a scale has eight notes.
The standard right-hand fingering for a C major scale is 1-2-3, 1-2-3-4-5. You reach E with the middle finger, the thumb travels under it and lands on F, the whole hand shifts right, and the remaining four notes follow.
Fingers 1, 2, 3 take you to E; the thumb passes underneath to the marked F.
A good thumb-under is inaudible — the scale should be a straight line, not "three notes, a bump, five notes". The bump has two usual causes: the wrist lifting at the moment the thumb lands, or the thumb jumping across at the last instant instead of traveling early.
Start moving the thumb toward its key from the third note, not when its turn arrives.
Let the elbow ease outward slightly so the whole hand crosses, rather than forcing it with the fingers.
Keep the wrist level — a wrist that rises and falls during the pass is exactly the bump you can hear.
Practice it slowly enough that the seam disappears, then speed up from there.
Common questions
How do you pass the thumb under smoothly?
Prepare it early and keep everything else quiet: the thumb heads for its key while the earlier fingers are still playing, the wrist stays level and the elbow drifts slightly outward. Go slow enough that you cannot hear the join.
Where does the thumb cross in a C major scale?
Going up with the right hand, between E and F: the thumb passes under the middle finger. The left hand does the same going down, between A and G.