Adjustment of Status
The process of obtaining lawful permanent resident status in the United States without having to leave the United States to do so.  Generally this option is unavailable to many persons who entered the United States without inspection.  This should be distinguished from "Change of Status" which is generally used when changing from one nonimmigrant status to another.  
Immigrant Visas
Permission from a U.S. Consul to seek admission to the United States.  A visa is issued subsequent to establishing eligibility for admission on a permanent basis under the immigration and Nationality Act.  Immigrant Visas generally permit an alien to be admitted to the United States for permanent residency.  It is for the immigrant to apply for admission during this period, and they are generally valid for six months.
Non-Immigrant Visa
Is a document that signifies that a consular officer believes that the alien to whom the visa was issued is eligible to apply for admission in a particular nonimmigrant category.  These visas can be valid for anywhere from 30 days to 10 years.  Visas can be limited to only a single entry or can be valid for multiple entries while they're valid.  These should be distinguished from the authorized period of stay which is given on the I-94, Arrival Departure Record.  
Defense of Marriage Act (DOMA)
One of the most significant developments in 2013 was on June 13, 2013 when a section of the Defense of Marriage Act (DOMA) was declared unconstitutional by the United States Supreme Court.  This ruling meant that the federal government was forced to recognize the marriages of same sex couples.  One of the rights that this ruling affects is the right for a U.S. citizen to seek permanent resident status for a foreign born spouse.  the couple, however, has to be married in a state or a country that recognizes same sex marriages.  This right exists even if the couple is currently living in a state that does not recognize same sex marriages.  
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The process of acquiring citizenship.  Generally speaking, in order to qualify for naturalization, a person must first be a Lawful Permanent Resident (LPR).  The person applying must be at least 18 years old, have resided continuously in the United States for five years after permanent residency.  This requirement is reduced to three years if the person gained their residency status based on marriage to a U.S. Citizen.  This requirement is reduced to three years if the person gaind their permanent residency status based on a marriage to a U.S. Citizen.  It is necessary for the applicant to have a residency for at least three months prior to the application for citizenship in the state in which they are filing their petition.  It is also necessary that the person is physically present in the United States for at least half the residency period.  In addition, it is necessary to maintain a continuous residency in the United States during the naturalization application process.  There is also a requirement of good moral character and a minimal ability to read and write english as well as understand the basic history and principles of government of the United States.  
Deferred Action for Childhood Arrivals (DACA)
On June 15, 2012 President Obama announced Deferred Action for Childhood Arrivals, otherwise known as The Dream Act.  This allowed children whose parents brought them to the United States prior to them reaching age 16 to be entitled to certain immigration benefits.  The administration has been using "prosecutorial discretion" to not initiate or even terminate removal proceedings against certain foreign nationals who qualify.  In addition these applicants are entitled to an Employment Authorization Document (EAD) which will allow them to work in the United States for a period of two years.  Generally this document can be renewed.