Why does the Earth rotate around a relatively fixed axis instead of rotating in all directions?

Conservation of angular momentum. And because we live in three dimensional space. Rotation occur in planes. In 1D space, rotation cannot exist. In 2D space, there is only one place that anything can rotate in, and thus only two kinds of rotation- forwards and backwards. In 3D space, there are an infinite number of possible planes you can choose that cut through space at different angles, and so things can rotate in lots of different directions; once you've picked one, though, you only have 1 spatial dimension left, which is not enough to create another plane, so any given object can only rotate in one direction at a time. In 4D space, once you've picked your first plane, you still have 2 dimensions left over, and you can pick another, independent plane in which to allow a second independent rotation; so, in spaces of more than 3 dimensions, things can rotate in a whole bunch of directions all at once. 3D space is also convenient in that there is exactly one left over dimension, which means that every plane of rotation can be equivalently represented by an axis vector perpendicular to the plane. So, we now know that in 3D space any object can rotate around exactly one axis at any given instant in time. The combination of the direction of the axis and speed of rotation give what we call the angular momentum vector. Angular momentum is a conserved quantity, which means that in an isolated system, not acted on by outside torques, the angular momentum will always be the same. That includes both speed and direction. (Conservation of angular momentum, in turn, is a result of rotational invariance- i.e., the laws of physics don't care which way you're pointing, and everything still works the same if you turn and face a different direction.) Thus, the Earth can only spin around 1 axis at a time, and that axis will not change, unless something outside the Earth shoves on it really hard. It turns out that tidal interactions with other bodies in the solar system actually do produce external torques which alter the angular momentum vector of the Earth, and cause the direction of its axis to change very slowly from moment to moment. This is so slow, however, that you will never notice the effects in your lifetime, although it does cause the pole to point at a different star every few thousand years (so Polaris was not and will not always be the north star).