TITLE: What is the suspension used in the Freudenthal suspension theorem?
QUESTION [5 upvotes]: The theorem states:

The suspension map $\pi_{i}( S^{n})\rightarrow \pi_{i}(S^{n+1})$ is an isomorphism when $i<2n-1$ and a surjection when $i=2n-1$. In the case where $X$ is an $(n-1)$-connected CW complex, this holds for the suspension map $\pi_{i}(X)\rightarrow \pi_{i}(SX)$.

But what explicitly is this suspension map?

REPLY [6 votes]: I am writing this answer as a supplement to Bruno's excellent (and sufficient) answer to provide an alternative perspective.
Let $X$ be a topological space and let $SX$ denote the suspension of $X$. Note that $X$ is the union of the upper cone $C_{+}X$ and the lower cone $C_{-}X$ (I assume that you have seen this before so I won't be explicit with the notation but I can be so if you like).
The suspension map is defined as follows:
$\pi_{i}(X)\cong \pi_i(C_{+}X,X)\to \pi_i(SX,C_{-}X)\to \pi_{i+1}(SX)$
where the isomorphisms can be derived from the appropriate long exact sequences of homotopy groups of a pair (using the contractibility of the cones $C_{+}X$ and $C_{-}X$) and the middle map is induced by inclusion. (Of course, this is assuming you are familiar with the notion of relative homotopy groups.)
Hope this helps!