It is Crassula Ovata, as Randy said. Jades will back bud but I don't know if they'll do it on such an unhealthy plant. A lot of what you do depends on the look you are trying to achieve. If you want a more "natural" in-the-wild kind of look, prune each branch back to about the point that two of the branches look to have been cut. You could go as far as an inch or two above the soil and it would live though. If you are wanting more of a one trunk tree look, separate it out, find the larges stem, prune it back to the point that it is not bending anymore and let it go. You could actually do both. Take one of the stems and plant it separately and pot up the rest together. Jades are very forgiving and will survive anything but being too wet! You can prune the roots, but it is not necessary unless you are wanting to put it in a bonsai pot. Get a terracotta pot. IMO they are better than plastic for succulents. Rinse most to all of the old soil off and inspect the roots to make sure they are okay. Prune at this time if you feel it necessary. Pot it up in a very free draining mix (1/2 commercial cactus mix and 1/2 pumice, high fired clay, or unpainted fish tank gravel). I have personally started using coir with my succulents instead of peat based soils because it is not so hydrophobic after it dries completely between waterings. Make sure your pot is not too big. The shorter ones made for azaleas, I think, work best. Wait one week before watering and place in a sunny window.

That's not a bad idea. I don't read the bonsai forum much and forgot about it.

Very nice sized jade although had you potted up over the years, I think it would be much larger. Definitely prune and prune hard. Do it soon. In the dead heat of summer, they slow their growth a bit, but I don't think you get as hot there as we do down here in Texas. Prune all of the thinner branches that are hanging of the edge of the pot to the point just below where they curve. The bigger, thicker branches, I would prune back to about three or four inches above where they V. Read some threads on rooting those cuttings and you'll have several more plants. I would definitely repot and make sure all of your heaviest branches are upright. You can take the whole plant out and losen the soil. When mine is being a pain, I often set my hose to shower and wash it off. I do the same with all the new plants I bring home to get off all the heavy soil they're grown in. You may have to cut some of the roots to separate the stems some or to get those leaning to be upright. It's okay. New roots will grow. I have seen these really cool supports at Home Depot used to support taller growing flowers or pepper plants. The smallest is 18" and they go all way up to 36". They are green and simply a long metal post that goes as far as you need it into the soil with an open circle at the top that is slightly bendable so that you can open it to put around the stem of your plant. In the end, if you used these to stake your branches, you would probably have to cut them off, but I think they'd work great. You could also use regular stakes and just tie them to the plant. Just make sure you check your ties to prevent them from cutting in. When you repot, wait a week or so before watering so that the roots have a chance to heal. This avoids rot.

Likewise, I wouldn't cut again without a clear reason to do so (rot). Let it callus a few days, re-pot in a dry coarse mix, then water in 1 - 2 weeks time - dribble some water around the stem, maybe a half inch out, without soaking the mix or hitting the cutting directly. Josh