There are many ways to test and analyze DNA, but there are three particular kinds of DNA testing that are especially useful in genealogy:  Y-chromosome DNA (Y-DNA) testing, mitochondrial DNA (mtDNA) testing, and autosomal-DNA testing (a-DNA).  I will discuss the first two of these here.

The genealogical usefulness of the Y-DNA and mtDNA tests is based on some special aspects of human inheritance:

1)  that the Y chromosome is passed on only by the father,
2)  that mitochondria are passed on only by the mother, and
3)  that neither Y-DNA nor mtDNA engages in "recombination."

Our inheritance is passed to the next generation via genes composed of an extraordinary substance known as deoxyribonucleic acid or DNA.  Most of the genes coding for our inheritance exist inside the nucleus of our cells and are thus called, nuclear DNA.  But some of our genes exist outside the nucleus, in the body of the cell, in small organelles called mitochondria.  This DNA is thus called mitochondrial DNA — or mtDNA, for short.

Our nuclear DNA is in the form of genes gathered together as chromosomes, much like the beads on a string (at least, it's functionally practical for us to visualize them that way).  Humans have 46 chromosomes arranged in 23 pairs.  One chromosome of each pair came from our mother via the nucleus of her egg and the other came from our father via the nucleus of his sperm.  In contrast, because the mitochondria exist outside the nucleus, floating in the body of the cell, we inherit mitochondria only from our mother's egg.

Our genetic gender is determined by one special pair of chromosomes called the sex chromosomes.  Unlike our other 22 pairs of chromosomes, called autosomes, the two sex chromosomes are not equal in size.  One of them is normal in size, but the other is much shorter.  By convention, these two sex chromosomes have been labeled the X-chromosome, for the normal-sized one, and the Y-chromosome, for the short one.  If your pair of sex chromosomes is XX, you will be genetically female; if your pair of sex chromosomes is XY, you will be genetically male.

The male Y-chromosome contains genes mainly involved with sperm production, so it makes sense for the female not to have these in her genetic makeup.  It does, however, leave males with only one X-chromosome, instead of two, the downside of which is that males are prone to some "sex-linked" diseases, such as hemophilia.

Because your mother has two X-chromosomes and, thus, no Y-chromosome, you cannot inherit a Y-chromosome from her.  You can only receive a Y-chromosome from your father, in which case you will be a son.  If you inherit your father's X-chromosome, you will be female.  (This phenomenon has led to the statement that "the father determines the gender of the child," which is essentially true because the gender of the child depends on whether the sperm fertilizing the egg is carrying an X-chromosome or a Y-chromosome.)  The fact that only males have Y-chromosomes means only males can participate in Y-chromosome DNA studies.  But why pick the Y-chromosome?

Before a sperm and egg join, the number of chromosomes each carries must be cut in half, otherwise the amount of DNA in our cells would double with each generation!  This reduction of chromosome number takes place during the last cell division that produces the sperm or egg.  In this "reduction" division¹, the chromosomes in each pair first wind and twist around each other breaking into segments and trading pieces back and forth in a process called "crossing over" and "recombination," just before the chromosome pairs separate forming two gametes (two eggs or two sperm) from what was one cell.²  Therefore, the genes from our two parents' chromosomes are "mixed up" in the chromosomes we pass on to our children, and only half of each parent's ancestry is passed to each child.  Only identical children inherit the same mixture, otherwise, each child has a different mixture, which is the reason siblings can be so different.  This mixing provides the genetic diversity that makes life on Earth so capable of adapting, but it makes tracing ancestry via genetics extremely difficult.  There is, however, an exception to this "mixing up" process.

Because the X and Y chromosomes are so different in size, they do not engage in crossing over or recombination during cell division, so their genes are not mixed up.  This means that men pass their Y-chromosome on unchanged from father to son, down through the generations (except for some occasional and trivial mutations).  The relative constancy of the male Y-chromosome is the basis for using it as a tool in genealogy because we can identify males who have a common patrilineal ancestor by the similarity of their Y-chromosomes.  As it happens, chance mutations on the Y-chromosome happen just often enough to be useful to genealogists.

We can't use the female X-chromosome in the same way because, even though it doesn't engage in crossing over with the Y-chromosome, females have two of them:  one from their father and one from their mother, so the inheritance of X-chromosomes is not in a direct matrilineal line, but can zig-zag between father and mother.  This is where mtDNA testing comes in...

Both males and females have mitochondrial DNA because mitochondria are essential to life³, so both males and females can take an mtDNA test.  Still, both genders inherit their mtDNA only from their mother, because it's transmitted in the body of the egg, not in the nucleus of the egg or sperm.  Like the DNA in the Y-chromosome, the DNA in mitochondria is passed on unchanged (except for rare mutations), so mtDNA analysis can reveal ancestry on your matrilineal line.

Between the two tests — a Y-DNA test and an mtDNA test — only two of your many ancestral lines are being tested, that is, either an unbroken chain of male ancestors or an unbroken chain of female ancestors, and any one person has only one each of these lines.  All the rest of your ancestral lines zig and zag between males and females, so these lines cannot be revealed with simple Y-DNA or mtDNA testing.  To test all your ancestral lines, you need to do autosomal-DNA testing (e.g., the "FamilyFinder" test at FTDNA).   Analyzing a-DNA results is a far more complex process and has the added downside that it's most sensitive back only about five generations, though with extraordinary effort (and a little luck) some claim to have gotten back seven or eight generations using it.

In contrast, Y-DNA and mtDNA, while testing only patrilineal and matrilineal lines, have great depth and can be used to track ancestry back thousands of years.  For this reason, they're used in studies of ethnicity and paleoanthropology — far beyond the reach of paper genealogy.

Because males pass on their surname, while the surname of females changes with every generation, there can be surname projects only for males.  Technically, in fact, women have no surname of their own.  When they're born, they're given their father's surname; and, when they marry, they're given their husband's surname.  There is no name that follows the female line, so there can be no "name" projects for females. 

I'll use my own ancestry as examples demonstrating the reason for this "males only" restriction.  First, my maternal grandfather's patrilineal ancestry is:

Martin STRAUB
_Hans Adam STRAUB
__Johann Pieter STRAUB I
___Johann Pieter STRAUB II
____Andrew STRAUB
_____Andrew S. STRAUB
______George Washington STRAUB
_______Albert Roy STRAUB

In contrast, my own matrilineal ancestry is:

Mary NEWGATE
_Elizabeth TOWNSEND
__Abigail MERRIAM
___Abigail MARBLE
____Abigail HALE
_____Abigail CHENEY
______Philancia Cheney HOLT
_______Martha Abigail HOPKINS
________Ida BROWN
_________Nina WILLIS
__________Nina Gale STRAUB
___________Diana Gale MATTHIESEN

After test results are known, the most logical groups for a female to join are a regional one and a haplogroup-specific one.  In my case, the British Isles Project was the one to join because Mary NEWGATE was an immigrant to New England with a presumed origin in England.  I am mtDNA Haplogroup T1a ("Clan Tara"), as are all the women on my matrilineal line (above).  My father is mtDNA Haplogroup H11 ("Clan Helena"), as are all the women on his matrilineal line (below):

Delilah Sampson GAULDING
_Elizabeth LaVernia WILLIAMS
__Delilah Ann McCRAY
___Maude Irene THOMPSON
____Anna California ROSE
_____Arthur Robert MATTHIESEN

There are a few ways mtDNA testing can be genealogically useful.  For example, a non-match can be used to expose a hidden adoption in a group of purported siblings, so in that respect, it can solve a few kinds of more recent genealogical problems.

If you are getting an mtDNA test — which I recommend because, at the very least, it's a fun thing to do — by all means read Bryan Sykes's book, The Seven Daughters of Eve.  It hasn't been a bestseller for no reason!

Lastly, patronymic surnames (patronyms) can be a particular problem, especially for those with Danish ancestry because Danes were among the last in Europe to adopt surnames, and many of those surnames were patronyms (discussed on this page).  It was for this reason that I used my maternal grandfather, not my father, as an example (above) for a Y-chromosome DNA surname project (i.e., because my father is a Dane whose patronym is changing with every generation).  The result of such late surname adoption is that a great many biologically related Danes have different surnames.  Conversely, because those adopted surnames were often based on the father's given name, a great many Danes with the same surname are not closely related biologically.  In such cases, DNA projects are best organized on a regional basis, which was my reason for opening the Danish Demes regional DNA project.