Qualify is a verb. I qualified to compete in the Olympics. Eligible is an adjective. I am eligible to compete in the Olypics.

To qualify, you actively need to achieve something, to be elegable, you passively are able to get something. For example: I have qualified for the final by winning this match. I am elegable to vote because I am 18 years old.

As Jan said, you asked about two different parts of speech, so that difference is the simple answer. Assuming you were asking about the difference between _qualified_ and eligible, then they are quite similar, and the difference is a bit more subtle. Both mean you meet the requirements to do, or to take part in something, and can often be used interchangeably. 'Qualified', as Lazar said, usually suggests you have worked to achieve something, whereas 'eligible' suggests you met the requirements more easily. You would typically be eligible for something if you were the right age, or you were a citizen of a country, or had lived in a certain place for a certain amount of time. You would typically qualify for something by having passed an examination or having met some standard in a competition. That's why we call a degree a qualification. That's also why I would not say that someone was 'eligible' for the Olympics. There are probably eligibility criteria to go to the Olympics, but people need to qualify to go there, and they do that by meeting a certain standard in their chosen sport.