The rope is connected twenty something feet up (the rope was originally 40 feet long, and didn't quite hang to the ground when it was doubled up. Hanging it was a bit of an engineering feet. The tree is not climber friendly, the first branches being fifteen or more feet up, and our ladder goes no where near high enough. So, I tied an old lock to a piece of strong and through it over the branch. It took a few tries, but eventually I got the lock stuck in the branches of a neighboring tree. No amount of pulling got it down, and eventually the string broke. But, that did provide a "proof of concept".
Round two had me using a socket with string, this time, in pretty short order I got the string around the larger branch and slipped under it, and around a second branch. I tied the other end of the string to the rope I bought at a boating supply store (thick soft rope is not available in most hardware stores). I successfully pulled the rope up and around. The two ends met somewhere just over the ground. Then I tied a strong slip knot into the rope and secured it to the branch. To be safe I retied the socket and string to the end, and looped it a second time.
With the rope secure and ten feet or so lying on the ground I proceeded to make the seat. I bought a cheap stool from target ($12) and removed the disk shaped seat. Then I drilled a hole in the center (this took some work because I didn't have a 5/8ths drill) Eventually getting the hole big enough for the rope. I tied it with a simple knot at a height tested for the kids, and when we figured that, I tied it more securely, and tested it with my weight. After bouncing on it for a bit, (the branch was big enough to barely move under my weight) I cut off the excess rope and burned it clean.
It isn't the best place for a swing. We have that fence right there. And there are numerous small trees on one side (I trimmed those back a bit). But it seems to work for the level of swinging the kids are into now. As they get older, they can figure out how to make it work better for themselves.