General Definitions -
"Short Block" - Engine Block and Rotating Assembly. Cam and crankshaft should be timed, with timing chain/belt installed. May include oil pump in a loose package
"Long Block" - Engine Block, Rotating Assembly, and Cylinder Head. Cylinder Head is typically assembled. Typically includes oil pump (loose pack)
"Compleat Engine" - Gets fuzzy here, and depends on who you talk to. A "compleat" engine (also called a "crate" engine) is usually ready-to-install, with all sheetmetal pans (oil sump, valve cover) in place, and the the water pump is usually already installed as well. Typically, does not include accessory brackets (like alternator, aircon, power steering, and the like.) Ask on this one! Different vendors use different versions of "compleat," and "compleat" almost never includes the manifolds (especially on inline engines. Sometimes, you'll get the intake on a V-block engine - but it would be protected in transit by the cylinder heads, where an inline would not. You'll probably never get the exhaust manifolds with a "compleat" engine...) The flywheel/flexplate is not included.
"Total Engine" - I've heard this one used at a few junkyards and other salvage outfits - this is an engine that has been pulled, and still has all of the bracketry, bolt-ons, manifolds, &c still attached. I've only seen engines like this that were pulled from vehicles - either running or salvaged - and I'd probably only buy them as cores (with the thought of going through them myself anyhow.) The flywheel/flexplate will probably be the one that came out of the vehicle, and may not work for what you're going to put the engine into...
Anything beyond "long block" or "short block" should damn well be verified before you plonk down your dosh - so you know what you're buying. "Long block" and "short block" have become industry standard terms, and the definition of those is (nearly) universal.