This goes back to the time of the settlers, when a group of people would get together in the kitchen, as this was the biggest room in the house, with a fiddler perched out of the way somewhere. The dances were well-known or easy to learn, and there wasn't usually a caller — the fiddler or one of the dancers reminded people if necessary.
In this form you spend a lot of time standing out, so for a busier dance you have both head couples, 1 and 3, leading the figure at the same time. Then both side couples lead it. But this would mean that each person only danced the figure once instead of six times. One way round this was to change partners. The head men led the figure twice, but with different ladies, so at this point all the ladies had danced the figure. Then the side men led the figure twice (probably helped by the ladies). You can also increase the activity level by having all four couples involved in the figure, though it's still usually led by one couple or one pair of couples.
There's also a difference between New England Squares (sometimes known as Quadrilles) which are phrased to the music, and Southern Squares which are unphrased because the music itself does not have strong phrasing.
Modern Western Square (MWSD) developed from this but went much further. There were hundreds of new figures, and you had to go to a series of classes to learn what to do at each level. As time went on the number of figures kept growing and the number of classes kept growing too. Now MWSD is in bad trouble, and some groups are trying to go back to the original idea of being able to enjoy the squares without so much teaching beforehand. There are also stylistic differences. In MWSD an allemande is a forearm turn; in traditional squares it is a hand turn. In MWSD a swing is walked and is quite a minimal move — just once around or even half around — in traditional squares as in contras and traditional English dances it's a pivot or buzz step.
If you're in a square formation make sure you keep the set square — don't drift away from your partner into some other formation. And don't let it get too big — that's often the reason people have trouble with squares. Make sure you know where to look for your partner and your corner — and be aware that the people may change each turn of the figure — know who your partner and corner are at any time. You may also hear about your opposite and your right-hand lady — though never your left-hand man, since traditionally the instructions are given to the man.
The step is a dance-walk — nothing violent, but not a plod or shuffle. Stand straight up, and look at people, but nothing artificial. I see some people in England dancing a circle with a skip-change step, but Americans would never do this; it's a purposeful walk.
The Figure to the dance is always walked through with the heads leading it (so get in head position if you're not sure of yourself). If it's a square where each couple in turn leads the figure, get into first couple's place. If it's a complicated figure it will probably be walked through for the sides as well. The Chorus or Break often isn't walked through, so you may get confused. The golden rules are:
It's always useful to know where you're supposed to finish at the end of each turn of the figure. There may not be a progression at all, as in the visiting couple squares and most MWSD apart from the singing calls. If you're changing partner, the figure will almost always be led twice by the two head couples working together, then twice by the two side couples. Usually the men have it easy — they finish in the same home place each time. Usually the women progress one place each time — to the right or the left. So women (and men) should know who their partner ought to be next time. Grab the right one. Failing this — at least grab someone. I often see men standing around disconsolately when I say “Swing the one you meet” — they didn't meet anybody. But I guarantee that there's a woman somewhere — preferably in your square — who is also not swinging anyone. Go and get her! Be positive. So what if it's not the right one — you can still keep going and enjoy the dance. It happens to all of us sometimes. But apparently some people (in both England and the States) prefer to argue about who went wrong last time instead of doing it right this time. Would you rather continue dancing or just stand there for the rest of the dance saying “He took my partner”?
If the whole set collapses: just reform your square — the men in their home place, the ladies wherever they wish — and wait for the start of the next turn of the figure. And smile! — most people won't realise you went wrong.