To weigh is all about judging; you could be seeing how heavy something is or if it is a good choice. You could weigh the tomatoes you're buying or weigh your options when applying to college.
To weigh something can mean to physically assess its weight — is it 3 pounds? 40 kilos? 22 grams? But it's no accident that the symbol for justice is often a scale. That's because all of the facts of a case need to be weighed or evaluated for justice to be served. And hopefully the decision will be balanced and fair.