Sometimes, problems arise when trying to extend continuous functions from $\mathbb{R}$ to $\mathbb{C}$. For example, consider the complex-valued function $f(z) = \sqrt{z}$. Consider the following (poorly-drawn) diagram:

If your domain $D$ sweeps out a quarter-circle of radius $r$, the the image $f(D)$ sweeps out half the angle (an eighth-circle) with radius $\sqrt{r}$. Suppose then that your domain $D$ was an entire circle in the domain.

Consider the blue-red interface in the domain. As the image suggests, $\sqrt{z}$ is discontinuous there, because as you transition across the line $x = 0$ in the domain, you jump half-way across the circle in the image. To deal with this, we extend $\sqrt{z}$ to a new domain: two complex planes "glued" together. It looks something like this.

Now the function is continuous. Instead of transitioning directly from (1) to (2) in the image plane, we "move through the cut" to the blue (3) on the second complex plane, so that $\sqrt{z}$ is continuous. Then we travel around the second plane, jumping from the red (4) to the red (2) in the first complex. We have gone around twice in the domain, and once in the image, and our issues with continuity are resolved. To be more precise about the gluing, the domain actually looks like this:

Now, as you can expect, you need to glue three complex planes together to make $\sqrt[3]{z}$ continuous, and so on, etc. Unfortunately, for the complex logarithm, no number of complex planes in the domain will make the image continuous. Thus, you need to define the complex logarithm on an infinite collection of complex planes, "spiraling" away in either direction. It looks like this.
Recall the example with the square root earlier. We had two copies of the complex plane, and their images under $\sqrt{z}$ were different. In the first copy, the image was the right semi-circle, and in the second copy, it was the left semicircle. Thus, when working with the complex square root, one needs to specify which copy of the complex plane one wishes to work with. Each such copy is called a branch (symmetrically, you can think of it as two copies of $\sqrt{z}$ defined on the same complex plane). Similarly, when working with the complex log, you need to talk about which of the infinitely many complex planes in the domain you wish to work with, and so you must specify which branch you are using. The popular choice is the so-called "principal branch":
Log$(z) = ln|z| + i$Arg$(z)$.