US systems check your name between agencies: CBP, USCIS, Social Security, the DMV, your bank and your employer. If the name is a little different in each place, you can get delays, a rejected license, or a tax mismatch.
One name only. Some passports have only one name. US systems need a first and last name. Your I-94 or visa may show FNU ("first name unknown") as the first name, or LNU ("last name unknown"). This is normal.
Name order. In many countries the family name comes first. US forms put the given name first and the family name (last name, surname) last.
Long names that are cut short by computer systems.
Different spellings between your passport and older papers.
Your passport is the master. Use exactly the same spelling and order on every US form.
Write your family name in the "Last name" box and all other names in "First name" and "Middle name", exactly as the passport splits them.
If you have one name, follow how CBP wrote it on the I-94. Many people with one name write it as the last name and FNU as the first name. Use the same choice everywhere.
When a system cuts your name short, write the full name on paper forms and keep a printout of the short version.
I-94 wrong: ask CBP Deferred Inspection. See Fixing a mistake on your I-94.
Social Security card wrong: ask SSA to correct it, with your passport.
Student record wrong: tell your DSO. They fix SEVIS and the I-20.
USCIS card wrong because of a USCIS error: ask for a free correction.
If you changed your name at marriage, carry the marriage certificate (with a translation). Decide which name you will use in the US and use it everywhere. Changing your name later means updating every record.
Adding a nickname or English name on official forms.
Swapping first and last name on one form only.
Letting a small spelling error stay. Fix it early.