Mathematicians picturesquely call the Poincare--Brouwer Theorem the "Hairy Ball Theorem." The picture derives from considering the Earth as a sphere with a hair growing out of each point on it. A barber has the task of combing the sphere so each follicle changes direction smoothly over the entire surface of the sphere. You can imagine a barber taking one of two strategies.

First, he might comb the hair from top to bottom leaving all of the hair on the top of the sphere pointing away from the North Pole and all of the hair on the bottom pointing toward the South Pole. The North Pole and South Pole are critical points (here the hairs undergo a sudden reversal of direction).

[figure depicting the first hairy ball]

Second, he might comb the hair around the sphere (like the jet stream circles the Earth). Here, too, the North and South Poles are critical points; there is no way to assign a follicle direction at these points that fits in continuously with the rest of the follicle directions.

[figure depicting the second hairy ball]

There is no way to eliminate critical points on the sphere in a hair combing. Each critical point represents a discontinuity in the unit tangent vector field on the sphere; therefore, no continuous tangent vector field exists on the surface of a sphere.