Flushing oils
These are special compound oils that are very, very thin. They almost have the consistency of tap water when cold as well as hot. Typically they are 0W/20 oils. Don't ever drive with these oils in your engine - it won't last. (Caveat : some hybrid vehicles now require 0W20, so if you're a hybrid driver, check your owner's manual). Their purpose is for cleaning out all the gunk which builds up inside an engine. Note that Mobil1 0W40 is okay, because the '40' denotes that it's actually thick enough at temperature to work. 0W20 just doesn't get that viscous!
Do I need a flushing oil?
Unless there's something seriously wrong with your engine, like you've filled it with milk or shampoo, you really ought never to need a flushing oil. If you're transitioning from a mineral oil to a synthetic oil, likewise you probably don't need to flush the engine first.
If you do decide to do an oil flush, first drain your engine of all it's oil, but leave the old oil filter in place. Next fill it up with flushing oil and run it at a fast idle for about 20 minutes. Finally, drain all this off (and marvel at the crap that comes out with it), replace the oil filter, refill with a good synthetic oil and voila! Clean engine.
Of course, like most things nowadays, there's a condition attached when using flushing oils. In an old engine you really don't want to remove all the deposits. Some of these deposits help seal rings, lifters and even some of the flanges between the heads, covers, pan and the block, where the gaskets are thin. I have heard of engines with over 280,000km that worked fine, but when flushed it failed in a month because the blow-by past the scraper ring (now really clean) contaminated the oil and screwed the rod bearings.
Using Diesel oil for flushing
A question came up some time ago about using diesel-rated oils to flush out petrol engines. The idea was that because of the higher detergent levels in diesel engine oil, it might be a good cleaner / flusher for a non-diesel engine. Well most of the diesel oil specification oils can be used in old petrol engines for cleaning, but you want to use a low specification oil to ensure that you do not over clean your engine and lose compression for example. Generally speaking, an SAE 15W/40 diesel engine oil for about 500 miles might do the trick.