The main difference between the three theories is how each of them perceive international relations. The realist views each state as being out for itself and power between states as a zero-sum game; where one state gains, another state must lose. The true realist views international bodies as being essentially toothless to prevent conflict because a leader has a responsibility to his or her country to place it in the best position to prosper and be safe. The realist acts in his nation's own self-interest, without regard to how it would affect others in a group.

The liberal views international relations as a collection of relationships between states. Liberals view collective action as being a good way of binding states together in common goals such as economic prosperity. They see international relations as a positive sum game; both sides can win in an agreement between states. Liberals place a high priority on collective action taken between international bodies.

Constructivist theorists believe that nations relate to one another based on preconceived conceptions of each other. Perceptions can change with a different leader's perception of another country or a people's prejudices against another region. The international system can be calmer or more chaotic based on different people's views of a region or group.

The previous answers are already quite good, but I have a bit more to add that I think will draw them out in high relief.

Realists believe that nations are motivated by self-interest.

Liberals believe that nations are motivated by ideals.

Constructivists believe that nations are whatever we make them to be.

In a fundamental sense, the constructivists are obviously right; nations are entities we created (relatively recently!) by collective action, and we could make them differently if we all agreed to do so. The harshest criticism one can really make about constructivism is that it may be too vague to be useful; it is undeniably true.

But there are aspects of liberalism and even so-called "realism" (I hate that name; "rationalism" isn't terrible, but I think I'd go with something like "atomism" or "self-interest theory") which are true.

Many nations are in fact motivated by ideals and capable of cooperating with one another to make the world more like what they wish it to be. International unions like the UN and the European Union are a testament to this. Indeed, nations themselves had to be formed at some point, and were formed from smaller units such as city-states in a manner very similar to the liberal vision of the international system.

And yet many nations are often motivated by self-interest, and often act in ways contrary to their stated ideals in order to achieve military or economic advantages. Often the darkest hours of a nation are found in these events---the Vietnam War and the Iraq War both come to mind as very "self-interest" sorts of international action that are contrary to stated ideals.

Realism is a bit of an overly complimentary name, something like "practicalism" or "rationalism" might be more appropriate. Realists believe that international relations are like a play with a strong Machiavellian influence; every state is an entity, and is looking out for itself. These states act reasonably and logically, and therefore we might assume that under this system, change would be slow, since few states would be willing to take risks, cooperate or be altruistic. This makes sense because a state which does not look out for its best interests would be rapidly taken advantage of and eliminated by the more competitive ones. 

Liberalism isn't exactly the opposite of realism. It believes that a state should act based on what it wants to be true rather than what is true; for example, where a realist state would act conservatively to protect its interests, the liberal state would act in order to make its views and intentions clear, even if it lacked the power to do anything about it. Liberalism takes a much less competitive and paranoid view of the world than realism, and tends to emphasized relationships that force cooperation, such as international trade. Some have argued that states that act with great fidelity toward this system of thought do not start wars.

Constructivism is at the opposite end of the scale from Realism; it believes that ideas are the driving force rather than states. More specifically, all things are given meaning based on the terms and definitions that we use to understand them; identical actions in different times or places can have vastly different implications, meaning that the acts themselves are less important than the constructed message they are a part of. This is a good way of understanding the "why" of any international relationship, although some have criticized that it lacks teeth.

The most important difference between these three has to do with how they conceive of the international system.

Realism conceives of the international system as anarchy.  It believes that states must therefore always act in ways that will get them the most power.  In such a belief system, organizations like the United Nations are pointless because states will only do things that will increase their power.

Liberalism sees a much less anarchic world order.  It believes that international bodies like the UN can create an international system that is based more on cooperation than on competition.  More international bodies leads to a greater chance for peace.

Constructivism thinks that the international order is contingent upon how states perceive one another.  What is important is if states perceive one another as threats or as potential allies.  This determines how peaceful the international system can be.