Implied odds (and reverse implied odds) are based on the possibility of winning (or losing) more money later in the hand. They consider the situation after the next cards have been dealt and explain situations where things are better (or worse) than pot odds make them seem. Put another way, implied odds is the ratio between the amount you expect to win when you make your hand (more than what is in the pot) versus the amount it will cost to continue playing. In contrast, reverse implied odds is the ratio between the amount in the pot (what you win if your opponent does not make their hand) versus what it will cost you to play until the end of the hand. One of the major factors behind considering implied odds is how hidden your hand is (how uncertain your opponent is of your hand); another is the size of future bets. For the latter reason, implied odds become more important in no-limit and pot-limit games than in fixed-limit games.
As an example of implied odds, consider that at the turn there is $12 in the pot, it is $4 to call (pot odds 3-to-1), hitting your hand means you very likely will win, and additionally your opponent is likely play to the showdown. If you miss you will simply fold (costing $4). If you hit you can expect to make an extra bet of $4 from your opponent, winning $16 total so your implied pot odds are 4-to-1.
For reverse implied odds, consider that you have a strong hand but little chance of improving and your opponent has a chance of improving to a hand stronger than yours, or possibly already has a hand stronger than yours (they have been betting and you are not sure if they are bluffing) - essentially a situation where you are not certain that you have the best hand. Say it is the turn and there is $12 in the pot and it is $4 to call (pot odds 3-to-1). If your opponent has a weak hand or misses their card they may stop betting in which case you would only win $12 (it costs $4 to find out you are winning). Otherwise, you have committed to playing to the end of the hand in which case it would cost you $8 to find out you are losing (pot odds 3-to-2). There are many variations to this scenario. The essential idea is that reverse implied odds should be considered when you are not certain you have the best hand; it will cost more in future betting rounds to discover this.