One of the easiest ways to identify a differential is by removing the cover and looking at exactly what's going on inside.
An open differential should be the most obvious, since it's basically going to be a housing with (4) Very visible gears inside. A welded should be obvious as well, since it's an open diff that has the gears welded together.
As far as identifying a clutch type over say, a Helical or Viscous, it really helps to know what the seller claims it is. Between knowing what it's supposed to be and having an idea of what they look like, you should be able to figure it out. Here's some reference pictures:
S15 Helical, dead give away are the (4) helical gears spaced evenly around the case:
S13/S14 Viscous, it has a significantly rounder body than most other diffs:
Clutch Type Differentials vary by manufacturer, best thing to look for is the 'Cam' mechanism that allows the diff to lock on Accel and Decel - which also helps to ID if it's a 1, 1.5, or 2 - way differential.
Nismo 2-Way (older Style, clutches are kind of visible):
Cusco 2-way (See the cam?):
Because there's so much variety between manufactures, and even models made by the same manufacturer, it's probably best to ask the seller what the diff is supposed to be and do some quick research before you buy.