No, I'm not talking about 3.1... Today I rehung the first of the stuck windows in my house; there are about 5 in all, plus several that open and close but with a lot of resistance. Since I could not find actual, step by step instructions anywhere on the web or in any books I looked in1, I thought I would write them out here. These instructions are only for windows with weights inside the frame; and specifically for windows built just like mine. I'm sure there are many variations on old window design; if you're taking yours apart and run into something different, you're on your own.
The first thing is to remove the sash molding. This is the rounded piece running up and down the side of the window frame, which holds the lower sash in place. ("Sash" is the thing that has glass in it and slides up and down.) The way to take this off is, first run a utility knife along where the molding joins to the window frame, i.e. the middle of the concave curve there; this will score the paint so that it does not chip during the next step. Now drive a chisel into this joint (assuming the molding is nailed on, not screwed) to force the joint apart. Once it is apart, use a crowbar to detach the molding. If the molding is screwed on, you will need to unscrew it first; this will probably be a huge hassle as there will likely be multiple coats of paint on top of the screws. It is a good idea to take both moldings off; being lazy, I only did it on the side where the weight was off, and this worked alright.
Note: William Duffield, of the Woodcentral Hand Tools Forum, recommends using two 1" putty knives, instead of a utility knife and chisel. One of them should be sharpened at the end. You will be able to run the knives along the joint and find the nails, and only pry where there are nails. This should significantly reduce the damage done to the molding.
Now you need to get access to the inside of the window frame. The way this was set up on my window was, underneath the sash molding was a screw holding a segment of the frame in place; once I undid it that piece came away. I have no idea how standard this setup is. Inside the frame will be a weight with a chain attached to it; the other end of this chain is supposed to be attached to the sash.
Remove the chain from the weight -- this should be straightforward -- and thread it through the pulley at the top of the windowframe. Your pulley may be stuck in place by old layers of paint -- mine was -- I do not think this is a big deal, the chain will just slide over the wheel instead of turning the wheel. Thread a nail or paperclip through the end of the chain, so you can leave it alone at the top of the pulley and work with the other end. Go back to the access hole in the frame, and fish out the chain, and reattach it to the weight. Put the weight back inside the frame.
Now you need to put the chain back onto the sash. Examine the other side of the sash (assuming that the weight was only off on one side) to see how it is attached. The way mine was on, was it threaded through a hole in the wood and then hooked on to a spring which was larger than the hole; the spring sat in a mortise in the side of the sash. Once both sides have their chains attached, you can put the sash back in its track, and nail the moldings back on.2
What I did next may be totally unconventional, I really don't know. At this point, the window slid up and down but, like several others in my house, it did not slide easily. This is particularly a problem for Ellen, who can't get some of them to move at all. So I just coated the sash track with bowling alley wax, and slid it up and down until it moved easily. I don't know how long this will last, but if it is still moving easily in a couple of days, I am going to do the same for my other resistant windows. Otherwise I guess I need to try something else...
1 -- to tell the truth I found several step by step guides to rehanging a window; but all were extremely non-specific, along the lines of "1. Take off the moldings. 2. Take off the sash. 3. Take off the sash cords. 4. Put it all back together." -- with accompanying full-color pictures that added nothing to the written instructions.
2 -- Several books recommended reattaching the moldings with screws, so that it would be easier to do maintenance in the future; this strikes me as ass-backwards thinking. Everywhere in my house where there are screws on an exterior surface, they are covered with many layers of paint and hugely frustrating. (The screws which held the piece of the window frame in place above were underneath the sash molding, so not painted.)