Enquire and inquire both mean to ask or seek information about something. The corresponding nouns are enquiry and inquiry.
Basically there is no big difference in meaning between these words. It can be generally said that inquire is the common form in the USA and enquire in Britain.
However, there are writers – particularly in Britain – who make a slight distinction between the two forms of the word. They use enquire simply as a synonym of ask in a general sense. Inquire is used with the meaning of making a formal investigation.
My boss enquired about the passing away of our dog.
The police are making inquiries to find the owner of the abandoned car.
If you write American English, you could use inquire, but otherwise you need not worry about which form is correct. As always, choose one form and be consistent throughout your text!