Bids or Passes are required to be made in turn, starting with the Dealer and proceeding clockwise.
(a "Bid" is a call of a number and a suit - Bids, Doubles, Redoubles and Passes are together defined as "Calls")
Acceptance
If a player makes a Bid, Pass or Double out of turn then the next player has the option of accepting this bid/pass/double (unless it was an "inadmissable" Double or Redouble i.e. illegal in the context of the auction e.g. doubling partner's bid.)
If the next call is made before anyone has noticed, then the illegal call has been accepted, you cannot go back (unless it was an "inadmissable" Double or Redouble).
Revert to correct player
If the next player doesn't want to bid then the illegal bid/pass is cancelled, withdrawn, and the auction reverts to the correct person.
And now it depends on whether the call was a Pass or a Bid, or a Double, and whose legitimate turn it was to call.
A. If it was made at right-hand opponent's turn then the offender must, if legal, repeat the Call after their opponent has called. Or they may make a 'comparable call'. Otherwise their partner will have to Pass at their next turn.
The Director will explain what calls would be deemed comparable, it depends on the situation.
B. If it was made at partner's turn, partner can make any legal call (and must try to ignore the illegal call).
When it is offender's next turn to call they may make a comparable call to the Pass/Bid/Double, or else their partner will have to Pass at their next turn.
C. If it was made at left-hand opponent's turn to call, and if that player has not yet bid, then again the auction continues normally, and when it is the offender's turn they can make a comparable call if they can, or else their partner will have to Pass at their next turn.
If left-hand opponent has already bid then the illegal bid is a "change of call" and a different law applies.
Lead Restrictions
If the opposition pair become Declarer.
If the illegal withdrawn call is not repeated, or replaced with a comparable call, then when offender's partner first comes on lead Declarer can prohibit them from leading any one suit not specified legally by the offender during the auction (and continue with this prohibition if they retain the lead).
EXAMPLE (at offender's partner's turn)
North opens 1NT but South was Dealer.
Firstly, East is allowed to bid over 1NT and the auction continues normally. If East doesn't want to do this, the bidding reverts to South. South may bid normally, but they must make every effort to ignore the withdrawn 1NT bid.
If South passes and West passes, North must repeat their 1NT bid and the auction continues normally. South can bid a suit, Stayman, raise to 2NT, whatever they like.
If South bids say 1S, and West passes, North may make any legal bid over 1S. If this is not comparable to a 1NT opening bid then South must pass. 1NT is 12-14 points balanced, so North can easily bid 3NT and this is close enough to be comparable to 12-14 that we allow South to bid again, they may want to bid 4H or 4S.
If South passes and West bids 1H, North may make any legal bid over 1H. A 1NT overcall would be 16-18 points and not comparable to a 1NT opening bid, so South would have to pass for one round. Calling any suit would not be comparable to 1NT, nor would a Double which could be made on stronger hands than 14 points. So really South is going to have to pass for one round, but they can bid later.