What's the difference between a NTC and a NMC?
A no-trade clause means a player cannot be traded without his consent. Consent is not required for waivers for assignment to the minors.
A no-move clause means a player cannot be traded, waived for a claim by another team, or assigned to the minors without his consent. This does not protect the player from a buyout.
Limited and modified NTCs and NMCs simply mean the player's contract includes specific terms for the clauses, usually allowing the player to specify a no-trade list of undesireable teams or a trade list of desireable teams.
If a player waives a clause to accept a trade to a new team, it is rare he will do so without a promise that the acquiring team will continue to honour the clause. Brad Richards, when traded from Tampa Bay to Dallas in 2007-08, is an example of this. However, if the player is traded before a clause has kicked in, that clause is automatically ruled void.
It should also be noted that clauses can only cover what would have otherwise been unrestricted free agency years.
CBA reference: Section 11.8 (a & b) (P. 48)