Step 1: Start from the formula
Glycerol is $\text{C}_3\text{H}_8\text{O}_3$, which is the saturated three-carbon alkane with three OH groups replacing three H atoms.
Step 2: Build the name
Parent: propane. Principal group: -OH, three of them, one per carbon, so the locants are 1,2,3 and the ending is "triol".
Step 3: Eliminate
A "diol" name accounts for only two OH groups, so (A) and (B) are wrong. "Propene" would imply a $\text{C=C}$ bond, which is absent, so (D) is wrong.
Step 4: Answer
Propane-1,2,3-triol, option (C).
Final Answer:
Three OH groups on propane give propane-1,2,3-triol. This is option (C).
\[ \boxed{\text{(C) }\text{Propane-1,2,3-triol}} \]