Whereas most chemotherapeutic agents attack the DNA of cancer cells, by disrupting their replication or perhaps inducing breaks in the helix, paclitaxel (or taxol as it is commonly known) instead attacks the "microtubules" of cells. What is a microtubule? In general, when cells are dividing into two pieces, the DNA must be synthesized (for the new cell) and then distributed into the two dividing sides. How does the cell put a copy into the new cell, while leaving one in the old cell?
The science is more complicated than that. When DNA is made prior to division, an entire new copy is made by breaking apart the helix, and using each half as a template to generate two helices. Each DNA helix is now a composite of an old half and a new half! These two helices are usually stuck to each other. Now that there are two DNA helices, these must be distributed properly. The way the cell does it is by throwing out protein filaments onto the DNA, latching it onto the cell membrane, and then pulling it apart as the cells split asunder. So the two DNAs are literally pulled apart. The protein filaments are the "microtubules". The name is actually amusing and reminiscent.
So what does paclitaxel (or taxol) do? It basically sticks to the building blocks of the "microtubules". These building blocks (like lego blocks) are called tubulin. Taxol uniquely sticks to tubulin, preventing it from assembling into a protein filament for the division process. Like other cytostatic drugs, taxol has a negative effect on cell division, and therefore affects the fast-dividing cancer cells, in contrast to the slow-dividing normal cells which don't feel such a strong destructive effect from taxol because they are not always in the state of division.