A hyperplane is a subset of a Euclidean space of one less dimension than the whole space. As such, it is defined by one linear equation. In $\mathbb R^3$, the plane created by the $x$ and $y$ axes is one such, represented by $z=0$. The plane $x+y+z=0$ is another, tilted and going through the origin. Each side of such a plane is a half space. The same happens in higher dimensions. If the coordinates are $x_1, x_2, \ldots ,x_n$, there is a hyperplane $x_1=5$ which divides the space into two half spaces: one with all points that satisfy $x_1 \gt 5$ and one with the ones that have $x_1 \lt 5$. Similarly, there is another with $x_1+4x_2+3x_3 = 9$ which is inclined in those axes.