We can define $S^2\subseteq \mathbb{R}^3$ as the vanishing set of $f(x,y,z)=x^2+y^2+z^2-1$. By equator you likely mean the intersection of $S^2$ with $P=\\{(x,y,z)\in \mathbb{R}^3: z=0\\}$ (for instance). In this case, the equator is the set of points satisfying $x^2+y^2+z^2=1$ and $z=0$, i.e. $x^2+y^2=1$. So, the equator in this case is a copy of $S^1$, the circle.
In the case of $S^3\subseteq \mathbb{R}^4$, we write $S^3$ as the vanishing set of $F(w,x,y,z)=w^2+x^2+y^2+z^2-1$, and repeat the same reasoning. We probably want to define the equator as the intersection of $S^3$ with $P'=\\{(w,x,y,z): z=0\\}$, which is the set of points cut out by $w^2+x^2+y^2=1$, namely a copy of $S^2$.