The definition you've most likely encountered is the following:

For maps $W\xrightarrow{f} X \xrightarrow{g} Y \xrightarrow{h} Z$, such that adjacent maps compose to $0$, $g$ extends to a map on the cone $\overline{g}: C_f\rightarrow Y$, but then the composition $X\rightarrow C_f \rightarrow Y \rightarrow Z$ is equal to $h\circ g$, so homotopic to zero, so $h\circ\overline{g}$ extends over the cone of $X\rightarrow C_f$, which is homotopy equivalent to the suspension $\Sigma W$, giving you a map $\Sigma W\rightarrow Z$.

There's a dual procedure: Since $h\circ g $ is nullhomotopic, $g$ lifts to the homotopy fiber $F_h$ of $h$. Now $W\rightarrow X \rightarrow F_h \rightarrow Y$ is equal to $g\circ f$, so nullhomotopic. Therefore $f$ lifts over the homotopy fiber of $F_h\rightarrow Y$, which is $\Omega Z$.

The map $W\rightarrow \Omega Z$ constructed this way is adjoint to the map $\Sigma W \rightarrow Z$ given by the other construction, so they are equivalent.

The second method can nicely be used to interpret so-called secondary cohomology operations in terms of Toda brackets:

For a cohomology class $\alpha\in H^n(X; \mathbb{Z}/2)$, suppose $Sq^1 \alpha = 0$. We can represent $\alpha$ by a map $\alpha: X\rightarrow K(\mathbb{Z}/2,n)$. We can also represent $Sq^1$ by maps, namely from $K(\mathbb{Z}/2,n)\rightarrow K(\mathbb{Z}/2,n+1)$ (for any $n$, those are related by applications of the loopspace functor). This gives us maps

$$
X \xrightarrow{\alpha} K(\mathbb{Z}/2,n) \xrightarrow{Sq^1} K(\mathbb{Z}/2, n+1) \xrightarrow{Sq^1} K(\mathbb{Z}/2, n+2)
$$
where we can form a Toda bracket since $Sq^1 Sq^1 = 0$ and we assumed $Sq^1 \alpha = 0$.

By using that the Bockstein fits into a fibration sequence
$$
K(\mathbb{Z}/2, n) \rightarrow K(\mathbb{Z}/4, n) \rightarrow K(\mathbb{Z}/2, n)\xrightarrow{Sq^1} K(\mathbb{Z}/2, n+1)
$$
which is basically the definition of the Bockstein, you can explicitly identify all the spaces and maps in above procedure. Since all of these spaces represent cohomology in various coefficients (and the maps natural homomorphisms between those), you can interpret everything in terms of lifting cohomology classes to other coefficients. In particular, you can compute this. The result $\langle \alpha, Sq^1, Sq^1\rangle$ appears otherwise as second differential in the Bockstein spectral sequence!

There are other examples like this, for reference read the stuff about secondary cohomology operations in Mosher-Tangora's cohomology operations book. They don't explicitly state things in terms of Toda brackets there, but it's a fun exercise to reinterpret this stuff (for example, the celebrated Peterson-Stein formula is just the juggling relation).