I got someone’s leftover yarn from a charity shop. Only one skein has the band with the details on it, but even then it’s perhaps half-used, so I’m not sure what yardage is left.
How do I know what weight it is, and how much yardage I might have? Or is it simply a case of guessing by comparison with my other yarn and knitting test swatches?
Of course, worst case scenario, I’ve got some scraps to play with and try out new stitch patterns on 
Lucky you!
You can figure out the yarn’s weight by measuring its wpi - wraps per inch. Get a ruler, wrap the yarn around it for a couple of inches, then count how many times it went around in one inch. Excellent directions here - http://www.craftsy.com/blog/2013/11/determining-yarn-weights/ .
As to how much you have, you can weigh a yard (or 10) of it, then weigh the whole thing and calculate the yardage that way. So, if one yard weighs 2 grams and your whole ball is 20 grams, you have 10 yards. You need a pretty sensitive scale that can weigh tiny amounts (a food scale for example) to do this.
You can do a burn test to determine the fiber. Nice chart here explaining how to interpret what you see when you do the burn - http://www.fiber-images.com/Free_Things/Reference_Charts/free_reference_charts_fiber_content_guide.html If you want to confirm something is wool, you can do a bleach test as well. Basically, wool/animal fibers will dissolve in the bleach overnight.