This is a drawing of how a comet behaves as it passes by the sun. Click on image for full size version (40K GIF) Images provided courtesy of JPL

When comets are kicked out of the Oort Cloud, they begin a passage into the solar system, spinning and tumbling as they come. The trajectory which they acquire can be hyperbolic, parabolic, or elliptic in shape. Only if a comet takes an elliptic trajectory will it make repeated passes by the sun, as does Halley's comet every 76 years.

As the comet comes closer to the sun, near the region of space occupied by Mars it becomes warm enough for the comet to begin to evaporate. When evaporation begins, the coma and tail form.

This picture shows that, once the tails forms, it always points away from the sun. That means that the tail will point in different directions at different places in the comet's orbit.