DNA testing can help you discover relatives and can also help you verify your existing paper trail.
There are three core types of DNA tests for genealogy, and each test has a different focus. I'll summarize them briefly:
Y-DNA: Tests your direct paternal line (father's father's father etc). Only men can take this test, and it's primarily used for surname studies. ie, Does one Smith family share a patriarch with another? Can also provide your Y-DNA haplogroup, which is your branch on the human Y chromosome tree.
mtDNA: Tests your direct maternal line (mother's mother's mother etc). Both men and women can take this test, and it's used to determine if you're related to someone via both your direct maternal lines. Also provides your mtDNA haplogroup, which is your branch on the human mitochondrial tree.
Autosomal DNA: Covers all lines of your ancestry. Both men and women can take this test, and it can estimate the degree of relationship between you and another person. ie, Parent/Child, 2nd to 4th cousin, etc. Also provides a percentage breakdown of your biogeographical ancestry.
So each type of test has a genealogical component, where your DNA is compared against other people in the company's database, and you get a list of your matches. Each type of test also has a deep ancestry component (haplogroup or biogeographical percentage), which gives you broader information about your ancestry.
When selecting a DNA testing company, be sure to read very carefully about what their tests include, since not all companies offer both the genealogical and deep ancestry components for each test.
You may want to visit the International Society of Genetic Genealogy (ISOGG) website for additional information. ISOGG is a non-profit group which exists to promote and educate about genetic genealogy. There are various resources on their website, such as info for newbies, success stories, a wiki, newsletters and more.