The different channels in bootstrap always refer to a four-point function; for simplicity we can take a scalar primary $\phi$, so the object of interest is
$$\langle \phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4) \rangle.$$
Up to a scaling factor, this equals a function of two cross ratios $u$ and $v$, say $g(u,v)$. To analyze $g(u,v)$, we can pick any conformal frame we like. For now, we'll map $x_1,x_3,x_4$ to
$$x_1 = 0, \quad x_3 = (1,0,\ldots,0), \quad x_4 = \infty.$$
By a suitable rotation, we can always assume that $x_2$ lies on the plane spanned by the first two unit vectors. Let's parametrize this plane by a complex coordinate $z$ with conjugate $\bar{z}$, so $x_2 \equiv z$.
We now use the fact that the OPE $$\phi(y_1) \phi(y_2) \sim \sum_i c_{\phi \phi i} \mathcal{O}_k(y_2) $$ converges inside a correlator $\langle \phi(y_1) \phi(y_2) \ldots \rangle$ if there's a sphere separating $y_1$ and $y_2$ from all other operator insertions. This is a consequence of radial quantisation. Let's apply this to the four-point function
$$ \langle \phi(0) \phi(z) \phi(1) \phi(\infty) \rangle \sim g(z,\bar{z}).$$
In passing, we've changed coordinates from $u,v$ to $z,\bar{z}$.
If $|z| < 1$ both the OPEs $\phi(0) \phi(z)$ and $\phi(1) \phi(\infty)$ converge at the same time. This means that that the four-point function admits a conformal block expansion
$$ g(z,\bar{z}) \sim \sum_i c_{\phi \phi i}^2 G_i(z,\bar{z}) $$
where $G_i(z,\bar{z})$ is the conformal block corresponding to the primary operator $\mathcal{O}_i$. We can call this the s-channel.
Now consider a different case: the disk $|z-1| < 1$. In this case the OPEs $\phi(z)\phi(1)$ and $\phi(0)\phi(\infty)$ converge. This is a different double OPE expansion. However, it should give back the same four-point function. In a formula:
$$ g(z,\bar{z}) \sim \sum_i c_{\phi \phi i}^2 G_i(1-z,1-\bar{z}).$$
This is a different channel, let's call it the t-channel.
Finally you can consider a third channel, when $z$ is far away from 0 and 1.
The magic of the (numerical) conformal bootstrap relies on the fact that if $z$ lies in the intersection of the disks $|z|<1$ and $|z-1| < 1$, the s and t channels converge at the same time. This gives you constraints on the spectrum without knowing $g(z,\bar{z})$ explicitly.
I want to stress that there is no consistent naming scheme for the different channels. So don't focus too much on the name "s-channel" or "t-channel" since different authors may use different names; it should be clear from the context what they mean. Also the $z$ coordinate above is nothing special: there are infinitely many other ones (other conformal frames) you can use.
I want to end with the message that by now there are many sets of lecture notes, MSc and PhD theses discussing the bootstrap program. With a quick Google search you will find a lot of pedagogical material. There are many too many tiny details that cannot be explained in a single forum post.
Q&A (4871)
