To impugn means to call into question or attack as wrong. If your usually grumpy brother is suddenly nice and sweet, you'll impugn his motives if you're smart — he probably just wants something from you.
The root of impugn is the Latin pugnare which means "to fight," so when you impugn, you are fighting or attacking, but by saying that something is false or wrong. If a candidate has a record that cannot be impugned, his or her opponent might go for a character attack. If you take pride in your work, you will be especially insulted if someone impugns your professionalism or integrity.