Lady Justice (the Roman goddess of justice) is blindfolded because justice is supposed to be blind or at least objective. Regardless of your situation, your race, class, gender, religion, etc. ... justice is supposed to be accessible to you. You don't have to look far to figure out this is the ideal and not necessarily practiced. If O.J. Simpson were a poor man, he'd probably be in jail now. If Rubin "Hurricane" Carter were a rich man or a white man, we probably would not have spent a good part of his life incarcerated. Justice being blind is the ideal we want to achieve and we fail short often.
The scales in her left hand is for the balance between the plaintiff and the defendant. The Egyptian goddesses Maat and Isis were depicted carrying scales as well thought to represent truth and fairness.
In her right hand is a double edged sword which give her the power to administer punishment or exoneration. Without the sword, the rest is meaningless. A good example of justice without a sword is the Wold Court which has little jurisdiction to enforce their rulings.