The 14th Amendment was one of the Reconstruction Amendments, designed to end forever slavery and inequality in the Union. Part of the price for being readmitted to the Union for seceded states was to ratify the 13th, 14th, and 15th Amendments, usually via the influence of Freedmen, Carpetbaggers, and Scalawags in the new state governments. This amendment was the most influential of the three; whilst the 13th Amendment was important at the time, there have been few cases of slavery in the US since - similarly, the 15th Amendment was neatly skirted for around 100 years by Jim Crow laws.
However, the 14th Amendment was the key to shattering Jim Crow, the method by which Dred Scott v. Sanford was nullified, and most importantly, the mechanism by which the Bill of Rights was extended to every citizen under every government in the purview of the United States of America. Serious business.
The Amendment itself says:
Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
Section 2. Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed. But when the right to vote at any election for the choice of electors for President and Vice President of the United States, Representatives in Congress, the Executive and Judicial officers of a State, or the members of the Legislature thereof, is denied to any of the male inhabitants of such State, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime, the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such State.
Section 3. No person shall be a Senator or Representative in Congress, or elector of President and Vice President, or hold any office, civil or military, under the United States, or under any State, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any State legislature, or as an executive or judicial officer of any State, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof. But Congress may, by a vote of two-thirds of each House, remove such disability.
Section 4. The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned. But neither the United States nor any State shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.
Section 5. The Congress shall have power to enforce, by appropriate legislation, the provisions of this article.
As one might expect, the first section became the most important. The part of Section 1 known as the Citizenship Clause immediately the invalidated the terrible decision of Dred Scott v. Sanford, which had declared that no black person could be a citizen of the United States. It ensured that even slaved recently freed of bondage were (in theory) entitled to equal protection under the law. In recent years, it has meant that the children of illegal immigrants, born in the United States, are citizens, and has led to some contentious discussions of what to do with their parents when caught.
The Due Process Clause was primarily used to strike down labour laws in the United States, but by the 1930s, this use of the clause was repudiated. Now, the Due Process Clause has been most famously held to protect the right to privacy, such as in Roe v. Wade, which you may have heard of - also Lawrence v. Texas, which you should take a moment and read about. Also, the Due Process Clause has been held to force judges who may exhibit a conflict of interest to recuse themselves; this may become very important as the current federal government appeals the original decision of Hornbeck Offshore Services LLC v. Salazar, which overturned the federal government's moratorium on exploratory offshore drilling in the wake of the Deepwater Horizon oil disaster.
The Equal Protection Clause has again been the source of both consternation and difficulty, yet equality and the end of a horrid era in US history. The Supreme Court held first in Plessy v. Ferguson that the Equal Protection Clause was not violated, so long as governments extended "separate but equal" services, ushering in a century of segregation that would lead to bitter court battles, entrenched governmental racism, several Academy Award winning films, and the government deployment of soldiers. The Equal Protection Clause was successfully used by the Warren Court in Brown v. Board of Education and the many, many successor http://en.wikipedia.org/wiki/Boynton_v._Virginia to that ruling; it was also used, in conjunction with the Due Process Clause, to overturn anti-miscegenation laws in the USA during Loving v. Virginia. It's also used to overrule the most heinous of gerrymandering attempts (see League of United Latin American Citizens v. Perry).
Finally, there's Incorporation. This is the big reason why the 14th Amendment is the best amendment there is. Before the 14th Amendment, the Supreme Court had held in Barron v. Baltimore that the Bill of Rights (those free speech, trial by jury, etc. amendments) did not apply to the several states, but only to the Federal Government. Basically, that means that the US government could not stop journalists from reporting on secrets in government, but the State of Virginia and the City of Detroit sure can! Obviously, this was a serious limitation on liberty, and one that led to the repression of what we now consider to truly be universal rights, at least in the western world. However, starting with the Slaughter-House Cases, and continuing with the Civil Rights Cases, the Supreme Court has incorporated most of the Bill of Rights to the several states via the 14th Amendment, including the rights to freedom of speech, religion, press, and assembly. The First, Second, Fourth and Sixth Amendments have been fully incorporated, and parts of the Fifth and Eighth Amendments have been incorporated.
Without the 14th Amendment, the Several States of the Union could never have become the rather more national and powerful United States of America we all know (and some love) today.