The tangent of a function curve affects the interpolated values between the keys of an animation. Most animation controllers use fixed tangents to define the function curve at a key location. The best way to view function curves is in Track View - Curve Editor mode.
By default, 3ds Max assigns Auto tangents to new animation keys. This results in smooth, natural motion through keyframes.
On the other hand, you might want to add corners and sharp turns to animation when called for. The Linear controller uses discontinuous tangents that point directly at the preceding and following keys, producing an abrupt change in motion at that key.
The two large flyout buttons at the bottom of the Key Info dialog provide five different types of predefined tangent types, plus a sixth type, Spline, lets you create custom tangents.
If you highlight a key in Curve Editor and see a line running through it with dots at either end, this is the tangent; it's available only for Auto tangents (blue handles) and Spline tangents (black handles). The handle on the left side of the key is the incoming tangent, and the one on the right is the outgoing tangent. Using the flyout buttons on the Key Tangents toolbar in Curve Editor, you can assign different incoming and the outgoing tangent types to all highlighted keys.
The flyout buttons with small arrows on the left side assign the incoming tangent to highlighted keys, and those with small arrows on the right assign the outgoing tangent to the keys.