The following is mostly off the top of my head, so I am not sure how many of my own conclusions here are also reached by other theologians:
Justice and mercy are often conceived to be in opposition to each other. And yet the Church teaches that God is both infinitely just and infinitely merciful. How is this possible? How can He exercise mercy without acting contrary to Justice, or vice versa? Catholics also believe that God is simple, that He has no parts. All His attributes, though in our minds they are distinct from each other, are in reality one and the same, namely God. There is no composition of parts in Him. If He could act in mercy while suppressing justice, and the other way around, this would imply a certain composition of parts in Him. So what is the solution?
Let me take an example. Grace. God gives His graces of forgiveness to the repentant sinner. The sinner, because he is a sinner, does not deserve these graces. But God out of His mercy grants these graces, say, when the sinner receives the sacrament of penance. Now, it is good to note that mercy here provides something new for justice to address, rather than suppressing justice altogether. The Catholic Church teaches, contra Martin Luther, that grace actually cleanses man of the guilt of his sins. Luther famously taught that man was a "dunghill covered in snow," i.e. that man is inherently sinful and evil, but that God merely overlooks and pretends that man is good, in order that man can be saved. So man is really completely unworthy of salvation, but God admits him anyway. But the Catholic belief is that God's mercy and grace actually cleanses man of His sins, thereby making him worthy, or able to be worthy. Justice can then work off of this ground: for now, man really does deserve salvation, if he cooperates with the workings of grace. So mercy, in this instance, has not destroyed justice, but given it a new foundation from which to do its work. To me, this is an amazing mystery of our faith, which should be absolutely mind-boggling and completely astonishing to us as believers. We ought to love God all the more for it.
But anyway... that said... There is yet another problem. What I have just said doesn't seem to solve the whole thing. Why did God exercise mercy instead of justice in the first place, when the sinner was stained with his guilt? Does not God suppress his justice at that moment by giving the sinner the graces of forgiveness? Even if later justice is renewed and is worked out to its full, before the reconciliation of the sinner, he is still sinful and still deserving of damnation; so why did God not exercise his justice then?
At this point we ought to look at the definition of justice, and make some distinctions. Justice is the rendering to a man what he is due, or rather giving to him what belongs to him. Now, let us make a distinction here: something may belong to man in two ways (probably more, but for our purposes I'm looking at just these two): first, it may belong to man because by his own works he has merited it; second, it may belong to man simply because of his nature as a creature of God, Who, in fact, originally intended it for man in creating him. And so it is with salvation. In the smaller picture, we may or may not deserve salvation, depending on our own personal merits. But in the larger picture, salvation always belongs to us, because God created us with that intention. God wants us to have salvation, and so he has created us in such a way that we naturally tend towards it; hence, according to our nature, it does in fact belong to us. Thus, it is not contrary to justice whenever God brings a man, even a sinner, closer to Himself, in order to save him, for by doing so God is aiding man in the attainment of what belongs to him.
St. Thomas makes the claim, if I remember correctly, that justice and mercy are both in all of God's actions. This is a reasonable claim, since it would address the problem of Divine simplicity. See how this is yet more reasonable, given the above: in aiding the sinner to his salvation, God is acting according to His justice, by aiding the man to the attainment of what belongs to him according to his objective nature. God is also acting according to His mercy, by giving man the grace to attain salvation despite his prior subjective guilt and demerit of that salvation. So there's a kind of objective-subjective distinction here. In the objective sphere, God is acting according to justice by giving man what belongs to him by his nature; in the subjective sphere, God is acting according to mercy by giving man what he, as a subjective individual, had originally demerited. Justice and mercy are at work here, in the very same act, and not in any way contrary to each other; and this because they work in different spheres, in regard to man, namely the sphere of his objective nature, and that of his subjective individuality.
So that's a couple ways to reconcile Divine justice and mercy that I've thought up. There is probably much more that can be said, but this what I have for now.