English surnames are called last names in English speaking cultures because they are placed last, after an individual's given name. A last name is often an occupation like Carpenter, Mason, or Smith. Or it may be the name of a father like Anderson, Jackson, or Thomas. Cities, towns, or villages are a derivation of some surnames like Greenberg or Burlington. Sometimes physical land features are used, like Woods or Rivers.
Surnames are formed and given in many different ways. Each culture has its own unique set of rules for surnames. Spanish and Portuguese cultures usually have two last names, one from the mother and one from the father. In some Indian cultures, the surname is an indicator of caste or social status. In Afghanistan, surnames may be an indicator of the tribe or geographical region of a family. In the Congo, some last names reflect the season or situation in which a child is born.
In many cultures, wives do not take their husband's surname, as is common in the U.S. In many Asian cultures, women keep their own surname. Conversely, in Pakistan, wives often take their husband's first name as their last name. Different variations of surnames for female and male family members are typical in Slavic and Greek cultures.
Lastly, many cultures have only one name or mononym, and no cultural surname. Many immigrants are made to choose a last name when they arrive in the U.S. Often the surname is chosen as a tribute name to honor a relative, perhaps a grandfather. Sometimes it is chosen for the region of the world in which they lived. One of my colleagues chose not to adopt a last name and the university and professors had an extremely difficult time with record keeping.
Using the term last name is confusing for many cultures because they list their family name first. Surnames are listed first in many Asian and African countries. I teach that the last name is a family name, but in some cultures, each family member has a different last name. These cultures give each child a unique last name. This is incredibly challenging for schools and organizations that assume every family member has the same last name.
No matter how a surname is given/chosen/created, it becomes an important part of a person's identity. Learning to pronounce both the first and surname of an international student is important to their identity and feeling of acceptance. In the past, many students were given 'American' names to make it easier for teachers. Fortunately, this is a rare practice today. Teachers and those who interact with multicultural students should always learn to pronounce their students' names correctly. I admit some names require a bit of practice, but by learning how to say a student's name correctly you convey your commitment to them as a person and show respect and appreciation for their culture.
For more specifics of how to record student names consult: https://ies.ed.gov/ncee/rel/regions/northwest/pdf/REL_2016158.pdf