Definition 25.2:

An $n-$cell is any space that is homeomorphic to the closed unit ball $E^n$ in $\mathbb{R}^n$.

Thus the square $I^2$ is an example of a two cell and the hemisphere

$$\{(x_1,x_2,\dots,x_n)\in S^{n-1}/ x_{n}\geq 0\}$$

is an $n-1$ cell.