Both are used. The difference in ending represents a change in number.
studet is the third person singular form (present active indicative).
e.g. Ubi studet filius tuus? -> "Where does your son study?"
Marcus studet. -> "Marcus studies."
student is the third person plural form (present active indicative).
e.g. the give Duolingo sentence
Marcus et Livia student -> "Marcus and Livia study."