The obstruction to spin$^c$ is often denoted by $W_3(M) \in H^3(M,\mathbb{Z})$. It is obtained as follows: Let
$$
\beta \colon H^2(M,\mathbb{Z}/2\mathbb{Z}) \to H^3(M,\mathbb{Z})
$$
be the Bockstein homomorphism obtained from the short exact sequence
$$
0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0
$$
Then we have $W_3(M) = \beta(w_2(M))$, i.e. the class obtained by applying the Bockstein homomorphism to the second Stiefel-Whitney class. This agrees with the Dixmier-Douady class of the (stabilisation of the) complex Clifford bundle $\mathbb{C}l(M)$.

Geometrically you can do the following: There is an exact sequence
$$
1 \to U(1) \to Spin^c(n) \to SO(n) \to 1
$$
Let $M$ be an oriented $n$-manifold and let $\pi \colon P \to M$ be the oriented frame bundle. This is a principal $SO(n)$-bundle. Let
$$
P^{[2]} = \{ (p_1,p_2) \in P^2 \ | \ \pi(p_1) = \pi(p_2) \}
$$
Then we have a principal $U(1)$-bundle obtained as follows:
$$
L = \{ (p_1,p_2,g) \in P^{[2]} \times Spin^c\ | \ p_1\cdot q(g) = p_2 \}
$$
where $q \colon Spin^c \to SO(n)$ is the canonical homomorphism. If we identify this $U(1)$-bundle with the associated line bundle, we have the following isomorphism
$$
\mu \colon \pi_{12}^*L \otimes \pi_{23}^*L \to \pi_{13}^*L
$$
where $P^{[3]}$ is defined analogously to $P^{[2]}$ and $\pi_{ij} \colon P^{[3]} \to P^{[2]}$ is the projection to the $i$th and $j$th factor. This should be thought of as a multiplication and satisfies an associativity constraint over $P^{[4]}$. This structure is called the lifting bundle gerbe.

How do you get the class $W_3(M)$ from this?

Choose a open cover $U_i$ of $M$, such that there are sections $\kappa_i \colon U_i \to P$, let
$$
L_{ij} = (\kappa_i,\kappa_j)^*L .
$$
Choose trivialisations $\theta_{ij} \colon U_{ij} \times \mathbb{C} \to L_{ij}$ (we can always choose the cover, such that these exist), where $U_{ij} = U_i \cap U_j$. Over the triple intersections $U_{ijk}$ we now have two trivialisations of $L_{ik}$. One is
$\theta_{ik} $,
the other one is
$$
\mu_{ijk} \circ (\theta_{ij} \otimes \theta_{jk})
$$
where $\mu_{ijk}$ is the isomorphism $L_{ij} \otimes L_{jk} \to L_{ik}$ induced by $\mu$. The trivialisations differ by a continuous map
$$
\omega_{ijk} \colon U_{ijk} \to U(1)
$$
which turns out to be a Cech $2$-cocycle due to the associativity of the multiplication $\mu$. Hence, it represents an element
$$
[\omega] \in \check{H}^2(M,U(1)) \cong H^3(M,\mathbb{Z}) .
$$
This is the Spin$^c$ obstruction class.