The 14th Amendment

Understanding the Constitution and its amendments requires an  understanding of the proper use of the English language, as it was  spoken when the Constitution was written, as well as knowledge of  history predating that of the Constitution.
 

The 14th Amendment states:
 

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.

The first portion of the section is the citizenship "definition" and "guarantee", which we shall split into three parts:
 

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.

The first part reads:
 

All persons born or naturalized in the United States.... 

The meaning of this part is clear; a person must be born within a State  or US territory, or go through the naturalization process, to be  eligible for citizenship. Furthermore:
 

 ...and subject to the  jurisdiction thereof....

This part narrows the scope of the previous part to include only those  individuals who fall under the jurisdiction of the United States. For  births, jurisdiction arises from the father rather than location of  birth, and gives two possible outcomes:
 

1) A parent who is a citizen, and thus subject to the jurisdiction of  the United States. This is the case for the vast majority of people born  in the US and grants automatic citizenship.
 

2) A parent who is not a citizen, and thus not subject to the  jurisdiction of the United States. This ensures that those born to  foreign diplomats, officials, military, workers, those illegally within the US, etc do not gain automatic citizenship.
 

The third part ensures citizenship when the first two parts have been satisfied:
 

...are citizens of the United States and of the  State wherein they reside.

The second portion may be called the "privileges and immunities section" and reads:
 

No State shall make or enforce any law which  shall abridge the privileges or immunities of citizens of the United  States 

It is essential to know history which pre-dates the Constitution for  this portion. The privileges and immunities alluded to here are those  privileges which one State may deny or restrict to people who are  citizens of other States, in order to place its own citizens in a  superior economic or legal position. One State may not arbitrarily tax, seize or dispose of property, limit the free movement of, etc of citizens of other States to a higher degree than they do their own citizens. Simply,  a State must treat citizens of other States exactly as they would their own citizens under said State's laws.
 

The third portion may be called a "due process section" reads:
 

nor shall any State deprive any person of life, liberty, or  property, without due process of law

This portion makes it a requirement for a State to give due process to any person within its territory, regardless of citizenship, when said person engages in actions said State considers crimes.
 

The final portion may be called an "equal protection section":
 

nor deny to any person within its  jurisdiction the equal protection of the laws  

This portion simply requires States to apply its own laws equally to any person within the State's territory. Again, pre-US history helps here.