Step 1: Understanding the Concept:
According to Huckel's Rule, a compound is aromatic if it is cyclic, planar, fully conjugated, and contains $(4n+2)$ $\pi$ electrons where $n$ is an integer ($0, 1, 2, ...$).
Step 2: Detailed Explanation:
1. Cyclopropenyl cation: Cyclic, planar, conjugated. $2 \pi$ electrons. For $n=0$, $4(0)+2 = 2$. It is **Aromatic**.
2. Benzene: Cyclic, planar, conjugated. $6 \pi$ electrons. For $n=1$, $4(1)+2 = 6$. It is **Aromatic**.
3. Cyclopentadienyl anion: Cyclic, planar, conjugated. $6 \pi$ electrons (4 from double bonds + 2 from lone pair). For $n=1$, $4(1)+2 = 6$. It is **Aromatic**.
4. Cyclooctatetraene ($C_8H_8$): Cyclic, fully conjugated in its structural formula, has $8 \pi$ electrons. $8$ follows the $4n$ rule ($n=2$). If it were planar, it would be anti-aromatic. However, to avoid instability, it adopts a non-planar tub-shaped geometry. Being non-planar, it is **Non-aromatic**.
Step 3: Final Answer:
Cyclooctatetraene is not aromatic because it does not follow the $(4n+2)$ rule and is non-planar.