What is the penalty for bidding out of turn?
Here is a summary of the relevant rules that @Pieter Geerkens linked to. I am leaving out some of the more sophisticated points, for example, the fact that an artificial/conventional pass call is treated as a bid for the purposes of these rules. The guiding principle is not to punish the out-of-turn bidder, but to make it impossible for the offending team to benefit from the information improperly exchanged. That's why many of them refer to Law 23, which allows a TD to further adjust the score if it is determined that the offending team gained an advantage by the improper call.

In most cases, a bid on LHO's turn is treated as a change in call, which is treated separately. In practice this is a difficult mistake to make--you don't usually make two bids in immediate succession.

Pass out of Turn


If no player has bid yet, offender must pass at next opportunity.
After any player has bid, offender must pass at next opportunity; in addition, if offender passes at his partner's turn to call, offender's partner may not double or redouble.


Bid out of Turn


If offender bids on RHO's turn, and RHO passes, offender must make the same bid, but there is no other penalty.
If offender bids on RHO's turn, and RHO does not pass, offender may make any call. If the call is the same "denomination" as the original call (same suit or NT), offender's partner must pass at next opportunity. If the call is any other denomination, offender's partner must pass through the entire auction, and if offender's team defends, lead restrictions will go into effect (more on that in a bit).
To explain the differences above: If partner bids 1C, and I get excited and bid to 2C before RHO passes, no harm done. If partner bids 1C, and I hasten and bid 2C, and RHO says, wait! I wanted to bid 2H, and I say OK, fine, 3C, that's not a very big deal. But if I say, wait, based on opponent's 2H bid, I want to bid diamonds instead... that conveys a huge amount of information about my hand to my partner, and allowing partner to continue to bid would be unfair.
If offender bids on partner's turn, partner must pass through the entire auction, and lead restrictions may apply here as well.
Lead restrictions: If a specific suit is named, declarer can require the offender's partner to either lead, or not lead, that suit at the player's first opportunity. A prohibition to lead the suit lasts as long as that player keeps the lead. If no specific suit is named, the declarer can prohibit offender's partner from leading any one suit.


Double / Redouble out of Turn


In all cases, opponents have the choice to allow the double or redouble to stand. In this case, there is no penalty.
If offender doubles on partner's turn, partner must pass through the auction.
If offender doubles on RHO's turn, and RHO passes, the offender must make the same double or redouble, and offender's partner must pass at the next opportunity.
If offender doubles on RHO's turn, and RHO does not pass, the offender may make any bid, but offender's partner must pass through the auction.