Fourth suit forcing is surely the most important part of anybody's system, and fortunately you can sit down with anybody and have a fair idea of how to play it, as most things are completely standard. In this article we shall cover the basics of the convention, as well as describing some of the variations.
When the bidding starts (all sequences will be uncontested ones here) 1♥-1♠-2♣ there are many hands with no obvious bid available, for example a stronger hand with no diamond stop. It is common to play 2♦ as fourth suit forcing here, saying nothing about the diamond suit, but showing eleven (possibly ten) or more points, and asking partner to describe his hand further, maybe by showing a diamond stop. You might worry about what to do if you actually have diamonds in this sequence, but usually you can bid no trumps with a hand like that, as it is unlikely you'll want to play in the fourth suit.
We shall not concern ourselves too much with when to use the bid, but more with how to respond to it. The key thing is to know which responses are forcing and which are not. This is very simple when the fourth suit bid is at the three level-it is game forcing (and thus to use fourth suit at the three level you need enough values for game opposite partner's opening bid). But what about the one and two level? We assume the sequence is as above.
2♥ is non-forcing. This bid essentially says nothing about your hand other than that you have a minimum and no other sensible bid (and thus five hearts). For example, x AQJxx xxx KQxx.
2♠ is non-forcing. Maybe Kx Axxxx xx KQxx. (Since many people would raise spades at the first opportunity with three spades and a weak hand, this should only show two spades for many people.)
2NT is non-forcing, obviously showing a stopper. Maybe x AQxxx KJx Qxxx.
3♣ is non-forcing, showing at least five hearts and five clubs. Possibly xx AQJxx x KQxxx. It is worth noting that here we start entering the realms of controversy, as some people would play this as forcing to game. However, if a new partner springs this on you it is probably safe to assume it is weak as he wouldn't have the confidence to spring it on you with a good hand. If you wish to play it as game forcing, then on the hand given you would simply bid 2♥.
All other bids are game forcing. Thus opener can rebid one of his suits, support partner's suit, or bid no trumps, but otherwise has to have enough values to force to game. Thus 3♥, 3♠, 3NT, and 4♣ are all natural, though note that 3♥ shows six hearts and 3♠ shows three spades.
Raising the fourth suit (that is, bidding 3♦) is another controversial area. Most people play this as showing a good hand with no suitable bid. Kx AKxxx xx AKxx would be a good hand fot the bid. With a hand like this you have absolutely no other sensible bid available, and if you wish to play this bid to show anything else you should seriously consider what you are going to do with this sort of hand. However, some people like to play it as showing that suit as well, just in case there is a fit in the fourth suit (and for descriptive reasons), though these hands can usually content themselves with bidding no trumps. (And often the fourth suit fit can be discovered later if necessary.)
How does bidding proceed after the opener has responded to the fourth suit bid? This is easy. Any bid is game-forcing. Thus whenever you want to make a forcing bid in a suit, you can always use the fourth suit first and then bid the suit, in the safety that partner will not pass you. (It is possible to play some sequences as non-forcing, but one should discuss this with partner, and it is maybe better just to keep things simple.)
There are two other related sequences that are interesting. One is repeat fourth suit forcing. The thing I play with my partners is that if you reuse fourth suit, then it is natural if partner has bid no trumps, and otherwise artificial. Thus 1♥-1♠-2♣-2♦-2NT-3♦ is natural, whilst 1♥-1♠-2♣-2♦-2♥-3♦ is repeating the fourth suit, still unclear where to play, and with no sensible descriptive bid.
The other sequence of interest is the jump in the fourth suit. For example, 1♥-1♠-2♣-3♦. Some people like to play this as natural, showing at least 5-5 in the two suits, others as a splinter agreeing clubs. One of my partners likes this as artificially agreeing clubs, as he points out that you are not often going to have a splinter in the fourth suit, but will often want some way of agreeing partner's suit with a strong hand without bypassing 3NT. This treatment certainly has a lot of merit. Some people play it as a cue agreeing clubs-this has considerably less merit! So what should one assume if you are playing with a partner for the first time? Not clear, but if you play splinters with them then it is probably reasonable to assume that it's a splinter, otherwise some random cue agreeing clubs.