Step 1: Understanding the Concept:
An allylic alcohol is a compound where the hydroxyl ($-OH$) group is attached to an $sp^3$-hybridized carbon atom which is directly adjacent to a carbon-carbon double bond ($C=C-C-OH$).
Step 2: Detailed Explanation:
1. Option A: $C_6H_5CH_2OH$ is benzyl alcohol. The carbon is adjacent to an aromatic ring, not a simple $C=C$ chain.
2. Option B: $C_6H_5-CH=CH-C(CH_3)_2-OH$. Here, the $-OH$ is on an $sp^3$ carbon (the one with two methyl groups) which is directly attached to the $CH=CH$ double bond. This fits the definition of an allylic alcohol perfectly.
3. Option C: $CH_2=CH-OH$ is vinyl alcohol (vinylic, not allylic).
4. Option E: $CH_2=CH-CH_2-CH_2-OH$. The $-OH$ is two carbons away from the double bond (homoallylic).
Step 3: Final Answer:
The allylic alcohol is option (B).