Here's a link to a
Wikipedia Article About Fuel Oil which should give you a pretty clear understanding of our various fuel oils (mainly from a North American perspective). There is some slight misinformation in there, because while Fuel Oil #2 (Home Heating Oil) CAN be basically the same as #2 On-Road / Off-Road Diesel, aside from color, the requirements for Home Heating Oil are dramatically lower than those for on-road diesel #2; the upshot is that you can (practically, not legally) use Off-Road Diesel #2 in your vehicle, but Home Heating Oil will most likely cause problems in relatively short order due to its composition. Also, a link to a
Wiki Article On Diesel Fuel. Reasonably good info there as well.
As others have pointed out:
Diesel #2: Standard, On-Or-Off-Road Diesel Fuel.
Diesel #1: Traditionally used to be a blend of Diesel #2 with kerosene / jet fuel. There are different ways of doing this blending, and the agent may not be kerosene. The important point is that it has reduced lubricity, lower energy content, lower gel temp, and lower freezing point than diesel #2. Used for cold weather operation. In many cold-weather fleets and agricultural settings (Northern US and Canada) only Diesel #1 is kept around to simplify logistics chains.
Home Heating Oil: Either Diesel #2 or Diesel #1, with colored dye added, and with fewer environmental and functional regulations on its manufacture. Depending on your area and the refineries supplying that area, it may be virtually identical to Diesel #2; but more likely it will be identical to really old, really crappy Diesel #2 that will damage your vehicle. Burn it in furnaces, not in engines.
Edit: The issue of Low Sulphur Diesel (LSD) vs Ultra Low Sulphur Diesel (ULSD) is another one entirely - it separates types / manufactures of diesel fuels, so there is in fact LSD Diesel #1, LSD Diesel #2, ULSD Diesel #1, ULSD Diesel #2, etc. Only ULSD is presently legal in North America as an On-Road motive fuel - that is to say that if you are filling up at a legal and reputable fuel station, you're getting ULSD, so there's no issue with putting the wrong diesel fuel in your new truck or car.
Low Sulphur Diesel / LSD : 500 Parts Per Million / 0.0005% Sulphur
Ultra Low Sulphur Diesel / ULSD: 15 Parts Per Million / 0.000001% Sulphur
Hope this info is helpful!