Final answer:
The relationship where one male mates and stays with one female is called monogamous. Polygynous mating, where one male mates with multiple females, is exemplified by the greater sage grouse. An advantage of monogamy may include increased offspring survival due to parental care from both parents.
Step-by-step explanation:
When one male mates and stays with one female, the relationship is called monogamous. This is a mating system where one male and one female are paired for at least one breeding season, and in some species, these associations can last a lifetime. In contrast, a polyandrous mating system is where one female mates with many males - an example of which is not directly provided in your question.
Considering the greater sage grouse, an example of a mating system where one male mates with many females is named polygynous. In polygynous mating systems, males may compete for territories with valuable resources to attract females, such as in the case of the yellow-rumped honeyguide, or may display elaborate behaviors in communal courting areas known as leks, like those seen in sage grouse and prairie chickens.
An advantage of a monogamous relationship could be the male-assistance hypothesis, which suggests that males that remain with a female to help guard and rear their young will have more and healthier offspring. This is often seen in many bird populations where both the male and female provide significant parental care.