1. To ask questions about something: When Karl and Jason met each other after several years, the first thing they inquired about was how things were going with their families.

The two different spellings are both correct; however, normally the British prefer enquire; so, a person can be inquiring, or enquiring, about what time it is.

2. To try to discover the facts of a case: The police were inquiring the neighbors asking if they had heard or seen anything unusual on the streets last night when a burglary took place of a family's house while they were away on vacation.