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In classical mechanics, a constraint is a relation between coordinates and momenta (and possibly higher derivatives of the coordinates). In other words, a constraint is a restriction on the freedom of movement of a system of particles.
Types of constraint

First class constraints and second class constraints
Primary constraints, secondary constraints, tertiary constraints, quaternary constraints.
Holonomic constraints, also called integrable constraints, (depending on time and the coordinates but not on the momenta) and Non-holonomic constraints
Pfaffian constraints
Scleronomous constraints (not depending on time) and rheonomous constraints (depending on time).
Ideal constraints: those for which the work done by the constraint forces under a virtual displacement vanishes.

References

Goldstein, Herbert; Poole, Charles; Safko, John (2002), Classical Mechanics (3rd ed.), Addison-Wesley, ISBN 978-0-201-65702-9

Physics Encyclopedia