If you play 2/1 Game Force (or "GF except rebid"), you need to decide what you will do on a hand worth forcing to game over a 1-of-a-major opening but without a good suit. For example, over a 1S opening:
The standard method is to bid 2C, establishing the game force. However, if you do this then it can become difficult for opener to know whether you have a "real" club suit or not, which makes slam bidding harder. In fact, it may sometimes be necessary to make a two-over-one response in a three-card suit (change one of the clubs in the example above to a spade).
One attempt at solving this problem is to play a natural game-forcing 2NT response. Removing balanced hands like the one above means that 2-level suit responses virtually guarantee a 5-card suit. However, there are three problems with this idea. First of all, the 2NT response consumes a lot of space, making further exploration on those hands difficult. Secondly, the range of game-forcing hands which do not have a 5-card suit is very wide: they can be any strength from 12 HCP up, and include 4-4-4-1 distributions as well as balanced hands. It is not really practical to put these all into 2NT, and so the 2NT response does not completely solve the problem it was designed for. And thirdly, it means that you can't use the 2NT response for other purposes (it is more commonly played as showing a good raise of opener's suit, of course).
A much better idea is to put all game-forcing balanced hands into the 2C response. So, 2C becomes an artificial two-way bid: it shows either real clubs or a game-forcing balanced hand. In some situations the hand might contain only two clubs.
Obviously this clears up the other two-over-one responses, since they now promise a 5-card suit. You might think that there is still a problem with the 2C bid, since we are putting even more hands into it than in standard 2/1, but in fact by making it explicitly a two-way bid, it becomes much easier to distinguish in the subsequent auction when responder actually has genuine clubs.
To see how this works, let's assume for the moment that we will play natural continuations over the 2C response. Now, assuming that opener rebids at the 2-level, we will require that responder always rebids 2NT with the balanced hand. Any other bid will show an unbalanced hand, and thereby implies a decent club suit. So for example, with the hand from earlier,
we will respond 2C to 1S, intending to bid 2NT at our next turn, even if opener's rebid is 2H. In standard methods it would be more normal to raise a 2H rebid to three, but playing a two-way 2C response the direct raise should show the "natural" hand type and promise good clubs. Of course, there is no reason why you couldn't bid this way even if your 2C response was defined as natural, but turning it into a two-way bid makes everything seem much clearer, as well as improving the definition of the 2D response.
Notice how by rebidding 2NT we put ourselves in the same position as those people who play a natural game-forcing 2NT response, except that opener has had one extra chance to describe his hand. We have gained an entire round of bidding, which is extremely useful.
So far this has all been very simple. However, it is possible to add a vitually unlimited amount of artificiality to the continuations after 2C. There are a number of reasons why it helps to move away from natural bidding.
First of all, consider a natural auction such as 1S : 2C , 2H : 2NT. What do opener's bids mean now? If you play natural methods then 3H, 3S and 3NT have obvious meanings, but 3C and 3D are less clear - presumably these would show a fragment, but considering that these are such cheap bids the meaning is not terribly useful. Furthermore, neither partner's strength is well defined - it would be better to give some information about strength at this point. So it seems better to use artificial continuations, for example 3C showing a minimum hand with 3D as a further artificial asking bid after that. This sort of scheme can be used throughout the system. Once you start doing things like this, the system starts to look rather like a relay system. Indeed, the 2C response is sometimes called a "relay".
Next, you might decide that it is not always most efficient to use 2NT as the bid which shows a balanced hand. Particularly after a 2D response to 2C, it makes a lot of sense to bid 2H with most balanced hands and use 2NT for something else. Doing this makes it look even more like a relay system.
And finally, it is possible to do a lot better than natural responses to 2C. In particular, using opener's 2D rebid to show diamonds is not particularly efficient: you want to use 2D much more frequently than that, particularly when responder's 2H rebid shows the balanced hand. Also, opener's bids at the 3-level need to be better defined. A typical scheme might look something like this (played by Bocchi / Duboin):
2D = any minimum without 4 cards in the other major
2H = any hand with 4 cards in the other major
2S and above = better than minimum, without 4 cards in the other major
Again, this looks very much like a relay system, with the minimum and better-than-minimum hands being treated symmetrically (1M : 2C , 2D : 2H , 2S/2NT/3C show the same shapes as 1M : 2C , 2S/2NT/3C).
Putting all these things together, the resulting scheme can be extremely complicated. But this is an area of system which rewards a bit of hard work. Balanced game-forcing hands are quite frequent, and are very difficult to bid in standard systems. A relay-like scheme is ideal for them: relays work best when the asking hand is balanced and fairly strong.