Hugh Stewart in his excellent booklet “Elements of English Country Dance” says: “Give weight: This is a common (despairing) cry”. Some people never seem to come to terms with it. If you're in a square set and you circle left, the men's hands are underneath and the ladies' hands are on top. The men push upwards and the ladies push downwards, so that there's some tension in your arms — you can actually feel that there's someone there. It's not an affectation; it's not something that you do because it looks pretty — it's sheer mechanics. It enables you to apply a force to the other person and thereby move them — while they're doing the same to you. The way to get a good circle (walked or slipped) is that you all give a slight pull to the person behind you. Try it.
Again with a star or a turn, giving weight makes a lot of difference. Not only does it feel much better having someone there on the end of your arm, it enables you to turn it round quicker — which you sometimes need. If you're doing a right-hand turn in the standard 8 steps, you can get by without giving weight. But try turning 1½ in 8 steps. In a turn you don't have one person with the hand underneath and the other on top — they're side by side — but they're still pushing against each other and there's tension in the whole arm. Don't try to turn it into an arm-wrestling match though; you're not trying to force a submission! And don't have your arm ramrod-straight; you need a slight bend to allow for differences in timing, or you'll jar each other. A hand at the end of a straight arm is very unyielding.
Try to 8 bars of music: Do-si-do. Right-hand turn 1½. Now with 16 bars, do the same movements and then back left shoulder and hand.
By the way (for the Americans), in English a right- or left-hand turn uses a shake-hands hold, not an American allemande grip with hands up and thumbs linked. And in English a star is also a shake-hands hold with your diagonal opposite — it's a proper “hands-across” star.
Another situation for giving weight is a longways set where the ones lead down the middle, lead back, and hand-cast round the twos, sometimes called an assisted cast. The common phrase is “The twos help the ones round the corner”, and the way they help is by giving weight. If the ones foolishly take a full 8 steps to lead down, then turn and come back, they've actually got no steps left for the hand-cast — so the giving weight is vital. If the twos aren't literally helping, the ones would be better off doing an ordinary cast than having to drag the twos round with them. Try it.
A third example of giving weight is a swing. What you don't want to do, ladies, is to sink down so that the man is supporting you — that's giving him too much weight. And some ladies — possibly because they have large dominating husbands — lean right back, which is also very hard on the man. You have to lean back slightly. Men, your right hand is supporting and controlling the lady; that's why ballroom position is the way it is. If you're both good dancers, the lady should be able to let go with both hands and still get a fast swing. I'm willing to demonstrate this, but I'm not advocating it — particularly in America which is famous for its litigation — because there are other factors to consider — another couple might bump into you, or you might find a slippery bit of the floor, or you might even discover that you weren't quite as good as you thought you were!
With a poussette, callers may say “Man push, lady pull” — but if you actually do that there's no weight there. It's more a case of “Lady pull, man resist slightly” — again so that you can feel there's somebody there. Try it palm against palm to see whether you're really working as a couple. And if there's some weight there you're less likely to do the commonly-seen “pseudo-poussette” where one person pushes and the other just bends his or her arms rather than actually moving backwards.
I've also been asked to mention a ladies' chain. The two ladies are not just touching hands as they pass by; they're giving weight to each other to help their onward movement. Equally the men are assisting them round rather than just rotating themselves.
Another example of giving weight, courtesy of Cindy Harris, is the apparently simple “Long lines go forward and back”. She explains that the whole point of giving weight is to be able to sense where the person(s) you're connected to are going (and when) and to let them know where you are going (and when). In long lines, first rule is “no spaghetti elbows”. Not rigid either — more like the branch of a tree that both resists and gives. If you are properly connected like that, then the person on either side will feel that natural shift of weight as you step forward — and you will feel them. Everyone adjusts slightly, and ideally that connection passes up and down the entire line. There's no need to admonish anyone to straighten up those straggly lines — everyone will be pretty much together and the lines will naturally be more or less straight.
What Cindy is describing is more subtle than than the obvious “giving weight” in a swing or an allemande, but the principle is the same.