Cardinality
The set $C(\mathcal{X},\mathcal{Y})$ of all channels$^{1,2}$ $\Phi:L(\mathcal{X})\to L(\mathcal{Y})$ has uncountably infinitely many extreme points. To see this, first note that every pure state is an extreme point of the set of density matrices. Consequently, every channel that sends pure states to pure states is necessarily an extreme point of $C(\mathcal{X},\mathcal{Y})$. In particular, every unitary channel is an extreme point of $C(\mathcal{X},\mathcal{Y})$.
Characterization
Necessary and sufficient conditions for a channel to be an extreme point of the set$^4$ $C(\mathcal{X},\mathcal{Y})$ are provided by theorem $2.31$ on page $97$ in John Watrous' book The Theory of Quantum Information.
Theorem $2.31$ (Choi) Let $\mathcal{X}$ and $\mathcal{Y}$ be complex Euclidean spaces, let $\Phi\in C(\mathcal{X}, \mathcal{Y})$ be a channel, and let $\{A_a:a\in\Sigma\}\subset L(\mathcal{X},\mathcal{Y})$ be a linearly independent set of operators satisfying$^5$
$$
\Phi(X)=\sum_{a\in\Sigma}A_aXA_a^*\tag{2.174}
$$
for all $X\in L(\mathcal{X})$. The channel $\Phi$ is an extreme point of the set $C(\mathcal{X}, \mathcal{Y})$ if and only if the collection
$$
\{A_b^*A_a:(a,b)\in\Sigma\times\Sigma\}\subset L(\mathcal{X})\tag{2.175}
$$
of operators is linearly independent.
Using this theorem, we can find more extreme points of $C(\mathcal{X}, \mathcal{Y})$, such as the amplitude damping channel and this unital channel.
$^1$ A channel is a completely positive and trace-preserving linear map.
$^2$ $\mathcal{X}$ and $\mathcal{Y}$ denote the input and output Hilbert spaces$^3$ respectively.
$^3$ In quantum information science we almost always deal with finite-dimensional vector spaces. Every finite-dimensional vector space over a complete field is complete, so using the definition of Hilbert space is strictly speaking an overkill. John Watrous uses the term "complex Euclidean space" instead.
$^4$ C.f. definition $2.13$ on page $73$.
$^5$ Here, $A^*$ denotes the adjoint of $A$, see page $11$.
