Step 1: Structure
Crotonyl alcohol (But-2-en-1-ol) has the structure $CH_3-CH=CH-CH_2OH$.
Step 2: Analysis
In this structure, the $-OH$ group is attached to an $sp^3$ hybridized carbon atom which is next to a carbon-carbon double bond ($C=C$). This specific arrangement defines an allylic alcohol.
Step 3: Conclusion
Since the hydroxyl group is in the allylic position, it is an allylic alcohol.
Final Answer: (A)