I received an email from someone whose cousins got married. They had questions about this. 

"We have had a situation in our family of two people (cousins) that were allowed to marry and they are being judged harshly. It's been so bad they ran away together. The two family members who married are children from half-brothers. These brothers share the same mother but not same father. What kind of relationship is this for their kids?"

Answer:

The relationship between the two people who married in this family is “half first cousins.” Whether cousins are legally allowed to marry is guided by laws that are determined by each state (not federal laws). Some states allow cousins to marry and some do not. Culturally, there are varying levels of taboo against this kind of decision, and the taboo is stronger in some cultures than others.

Where the "half" comes into play in a relationship is a common point for confusion, so I'll explain that a little further. When a person in the grandparent generation has children by two people (if a grandfather has children by two wives, for example), then those children are half-siblings to one another. 

When half-siblings have children, those children are first cousins, but technically only "half" by DNA. They only share ONE grandparent (the grandfather) in common instead of TWO grandparents; this means children of half-siblings only have half the amount of DNA as is typical for first cousins.

There can also be double first cousin relationships and three-quarter sibling relationships. Other posts like this one on double first cousins can help you understand these better. A three-quarter sibling shares one parent in common and their other parent is a first degree relative to the other. For example, if the two wives are sisters to each other, then the half-siblings produced by the unions of their father with their mothers are more like "three-quarter" siblings. For three-quarter relationships, there are more ancestors in common than is typical for most straight forward "half" relationships.

Wondering if cousins can marry? An article in the online magazine Slate addresses this question and points out it is state by state. It also covers what the research and experts say about cousins about the medical impacts of being a child born to cousin couples. Long story short? There is not a significant difference for second cousins or more distant.

Readers of this post who are interested in more information on this topic might find the book The DNA Guide for Adoptees helpful.