Required usually implies a rule is involved. For example, being required to carry ID when in a government building.
It is also used in place of “needed”.
“If you have a bad toothache, a dentist is required.”

Obligatory is used in social settings, not really a legal binding one, or caused by a formal rule.
For example, it’s obligatory to visit the relatives every New Year.

Compulsory and mandatory are quite similar. The main difference is mandatory involves conditions, and is usually used in legally binding terms.

Compulsory just means it is essential, and cannot be postponed. It also shares the first definition of required. You will see “compulsory” mostly in schools and businesses.

“It is mandatory to state your name and birthday in the application form.”
“It is compulsory to wear uniform to school”