For $(X,x) as$ pointed topological space, then its reduced suspension $\Sigma X$ is equivalently
obtained from the standard cylinder $I\times X$ by identifying the subspace $(\{0,1\}\times X) \cup (I\times \{x\})$ to a point.
(Think of crushing the two ends of the cylinder and the line through the base point to a point.)
obtained from the bare suspension $S X$ of $X$ and identifying $\{x_0\} \times I$ with a single point.
obtained from the reduced cylinder by collapsing the two ends, i.e. the cofiber
the mapping cone in pointed topological spaces formed with respect to the reduced cylinder $X \wedge (I_+)$ of the map $X \to \ast$;
the smash product $S^1\wedge X$, of $X$ with the circle (based at some point) with $X$.
For CW-complexes the reduced suspension is weakly homotopy equivalent to the ordinary suspension: $\Sigma X \simeq S X$.
suspensions are H-cogroup objects
Up to homeomorphism, the reduced suspension of the $n$-sphere is the $(n+1)$-sphere
See at one-point compactification – Examples – Spheres for details.