1NT overcalls are standard takeout doubles with the exception that they cannot have any very big hands. The normal maximum is about 18 HCP with a void in opener's suit, 17 with a singleton, and 15 with a doubleton. With more, start with a power double.
NTOs guarantee at least three-card support for every unbid suit. Without that, don't do it. They promise 2 or fewer cards in opener's suit. Exceptions to that ought to be very rare. The reason for these stringent requirements is that advancer is going to take advantage of the limited nature of the 1NT overcall and jump preemptively a lot more than vs. a standard takeout double, because he doesn't have to worry about a strong jump overcall. Passes are very rare and tend to deny a four-card suit outside of opener's. Since advancer's hand must be weakish, almost all of the high cards are in that suit. He should not have more than 10 HCPs in general.
3-level bids are blocking, long suits, limited values. Except red on white, they are not invitational, although with a good fit and a good hand, NTOer can bid game. Expect about 5-6 HCP and a good 5-card suit, although it will be six fairly often. Red on white, jumps are not strictly invitational, either, but they are sound hands, so NTOer can move with a good fit.
All invitations start with the cue bid or 2NT. 2NT is a normal 2NT response (11-13ish) but with a better stop, since the opening lead will come through it.
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Jumps are preemptive. Think about jumping in any five-card suit if you have two or fewer cards in their suit; RHO is going to have to support partner, and if he takes a simple preference at the 3-level, that will sound ambiguous, and if he jumps to the 4-level, they've had a nullo auction. If advancer passes, opener redoubles with any five-card suit. Advancer responds in the cheapest suit he can tolerate playing vs. a five-bagger. After doubles, bids in opener's suit are natural.
If responder bids a new suit or raises opener,
doubles are responsive (takeout). At high levels
(4 and up) these doubles just show cards and
will be passed more often than not, although
(1m)-1NT!-(4m)-X is takeout for the majors and
is not passed often.