In the past, I've dealt with Constitutional legal issues, one of which was heard before the US Supreme Court, although I was not chief counsel.
First off, I would recommend reading through it if you haven't done so already. I know this may seem obvious, but I find that many people have never done this. A quick reading of it (it's not that difficult a document to read or that difficult to understand) will help give you an understanding of it.
Second, you have to understand a little of the background that precedes the creation and eventual adoption of the Constitution.
The first attempt at creating a new government after declaring independence was the Articles of Confederation. Inherent in this first attempt was a great disdain and distrust for the government. Consequently, they kept many powers out of the government, and the government suffered for it. It was unable to do many things, and it became readily apparent that this new government was ineffective.
When the time came to write the new Constitution, there two basic schools of thought. One school recognized that the Articles of Confederation created too weak a government to actually govern and survive. The new government under this new Constitution would have to be stronger and more capable. More capability, however, carried with it more power and authority.
The other broad school of thought also recognized that the Articles were weak, but felt that that was the lesser of two evils. It was better to have a weaker government than a stronger one that could potentially grow too strong.
So the Constitution was written as a compromise between these two basic camps. The Bill of Rights came later as a means to appease those who felt that the new Constitution created too strong a government. If you look closely at the Bill of Rights, they mostly provide clarification on the Constitution; the Ninth Amendment explicitly states the fact that any right not given to the government inherently rests with the people. Amendments that came later usually provide new rules or authorities.
Third, you have to understand what the Constitution is, who it's speaking to, and what it's saying.
The Constitution represents the framework of our (assuming you're American from here on out) government and, consequently, our society. It lays out the branches of government; it assigns basic roles, responsibilities, duties, and, perhaps most importantly, limitations.
The Constitution is a document directed at the government itself. It is not speaking to you or me. It is a set of rules the government must follow; it is not a set of rules the people must follow. If the Constitution doesn't allow the government to do it, or if the Constitution limits the government in a fashion, the government cannot do something.
For example, the First Amendment does not grant the citizens the freedom of speech. Rather, it prohibits the government from making any law restricting our free speech. In other words, we already have an inherent freedom of expression, and the government cannot restrict it. But private citizens can. Remember, the Constitution does not apply to, and is not directed towards, private individuals. This is why you can protest or picket outside a government office on government property, but a privately owned mall can kick you off the premises for doing the exact same thing with the exact same message.
The basic concept behind the Constitution is that originally any and all rights rest with the people. These rights are inherent and self-evident. The people, however, deemed it necessary to create a system to govern. In doing so, they understand that they must surrender some of these rights and privileges to make it work; likewise, this new government will need to slightly curtail some of their privileges in order to maximize the benefits to everyone. So, they created a document that governs the government. The government is only as powerful as the Constitution says it is. Any right, freedom, or privilege not given to the government stays with the people (Ninth Amendment). Remember that all rights originally lay with the people, so unless such right is explicitly given away, it remains with its natural owner, the citizen.
The Constitution represents inherent fears and distrusts that the founding fathers had with the idea of government. They had just thrown off the governance of a system that invested a high amount of control in one individual (the monarch) and a body of officials with limited authority and oversight of the monarch. This system did not allow for direct representation of all citizens, and many in the Americas felt that it allowed for arbitrary measures to be taken without any opportunity for the affected to be heard, seen, or known.
This is why when the Constitution was written the very first portion of it, after the preamble, deals with the formation of government. And the very first body of government to be identified is the Congress. Article I details the Congress. The president doesn't come until Article II; this is a small detail which serves to demonstrate their reluctance to have a strong executive branch.
Reflected in the Constitution, as well, are various ideals. The very first words are, "We the People." It is a new government founded by the people, for the people, and to be composed of the people. There is no royalty or nobility who head the government or who receive special privileges. (Obviously, this is laughable to some extent given the fact the women and non-whites were treated as anything but "the people.")
The first three articles of the Constitution deal with the formation of the government. They give these three branches various duties and authorities. Importantly, though, these articles are silent on a great deal. Since the government cannot assume a role not enumerated in the Constitution, silence in the Constitution means that no authority is granted. However, the limited authorities that are granted are broad and somewhat vague. The importance in this, and arguably the genius in this, is that it allows for the government to adapt and reform itself to suit the times.
The sections in Article I, for example, are a good example. They range from very specific duties and requirements for the Congress, but they also deal with broad powers. For example, because the founding fathers were so anxious about one branch of the government gaining too much power, they instituted the checks and balances system. Furthermore, they detailed a bicameral system. One house checks the other; one house gives weight to population; the other other house treats all states as equal. One branch is always checked and subject to approval on actions from another branch. (Congress passes a law. The President must approve it, but Congress, if properly motivated, can override the veto if they wish. Any approved law, no matter how passed, is subject to review by the Courts.)
On the other hand, regarding the broad and somewhat vague provisions, they allow Congress the leeway to meet unforeseen demands and situations. Article I, Section 8 gives Congress the power to pass laws that "provide for the common defense and general welfare of the United States." This is pretty broad, right? That's the way they intended it. The genius of the founding fathers' actions is not that they thought of everything when they created this new government and country; their genius lies in the humility they showed in recognizing that they didn't and couldn't think of everything. So they built in some wiggle room to the document and to the government. But they built in limitations to this wiggle room. Section 8 further defines what Congress can do. Essentially, any measures that must be taken that deal with interstate or international matters are the duty of Congress and no one else.
What the founding fathers did was to create a system that was flexible; it is limited yet strong. It has the ability to meet the changing needs of a dynamic country and the changing times. It gives the government various duties to perform, and it, importantly, gives the government the power to perform those functions while simultaneously keeping the power vested in the people.
The citizens have the authority and the ability to replace, remove, or otherwise deal with representatives that are not representing the constituents properly.
The parts of the government not immediately answerable to the people check themselves against other branches of the government.
The Constitution is the framework for this. Embedded in that document is the genius of countless people from the formation of the union to today. The hopes, fears, aspirations, and anxieties of its writers are evident in the system they created. It's withstood the test of time; Civil War, emancipation, assassinations, impeachments, and countless other events have proven that the document works and the government it set up is stable capable of moving forward into the future.
Billy Rufus -- Thank you for the Reddit Gold!