Shal, "not keeping the mitzvot" is a pretty good description of sin. But there are degrees of sin.

"Missing the mark" is one of the sins most often commited. It is a chatat -- which does come from the Hebrew word for an archer missing his mark.

Then there is avon which is an act you do impulsively -- you just can't stop yourself even though you know it is wrong.

A pesha is one of the worst types of sins. This is a sin you do knowingly, and against G-d. Murder would be a pesha where manslaughter is an avon.

There are a few others, but those are the biggies.

But buttom line, your definition is right. Sinning is doing something that takes you farther away from HaShem -- which would be by not doing those things that draw you nearer to Him (mitzvot).