the cathode attracts cations because thats where reduction takes place, and the anode attracts anions because thats where oxidation takes place...
in a galvanic cell, yes the anode is negative. But when you're trying to run a gel, for like proteins or DNA, you're making an electrolytic cell, in which the anode is actually positive. Therefore, the negative DNA will move toward the anode
How I think of a gel is as a salt bridge (which is probably wrong, but I'm pretty sure it works). Electrons flow from anode to cathode, so to balance charge, negatively charged particles are drawn to the anode.
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