Both of these verbs are used for asking about something or looking for information. The nouns for the request / investigation are enquiry and inquiry.
Technically, enquire is used for "ask" in general ("He enquired about prices for the products"), and inquire for a more formal investigation ("The police will make an inquiry into the alleged corruption.")
However, in American English it is now very common to use inquire for everything: "He inquired about prices for the products."
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