Income from self-employment is off-set against unemployment benefits. Does this mean income from self-employement terminates your unemployement, or they take the difference away from your unemployeement check

I'm guessing it means "they" work out the difference.

Ok, first each state has their own laws and rules on unemployment.

In GA, you are allowed to earn 50 dollars before it starts to effect your unemployment check, then after 50 dollars for each dollar earned, it takes away a dollar of unemploymnet.

Off set is to take the place of it. So you still get your unemployment but it will be reduced by the amount of your earnings.