Suppose you have a mineral sample that scratches fluorite but not topaz. What can you tell about?

Well, you'd look up the hardnesses on the Mohs scale. I now fluorite is 4 but I don't know what topaz is so I'm going to have to go and look it up. Okay, apparently topaz is 8.

The mineral you have doesn't scratch topaz so its Mohs hardness is less than 8 (i.e. it's not a diamond). It does scratch fluorite so its hardness is higher than 4 (i.e. it's not gypsum).

The things it could be (again, I'm going to have to look this up. I should know this stuff but I seem to have forgotten) are:

4–4.5 platinum, steel
5 apatite, cobalt, zirconium, palladium, tooth enamel, obsidian (volcanic glass)
5.5 beryllium, molybdenum, hafnium
6 orthoclase, titanium, manganese, germanium, niobium, rhodium, uranium
6–7 glass, fused quartz, iron pyrite, silicon, ruthenium, iridium, tantalum, opal
7 osmium, quartz, rhenium, vanadium
7.5–8 emerald, hardened steel, tungsten, spinel

That last bit was from Wikipedia.