These are two spellings of the same word, which means to seek information about something or to conduct a formal investigation (usually when followed by “into”). The corresponding noun is enquiry or inquiry.

Either spelling can be used, but many people prefer enquire and enquiry for the general sense of “ask”, and inquire and inquiry for a formal investigation:

• I enquired his name
• The first enquiry in my inbox today was about lost property.
• We are going to inquire into the incident.
• The lawyers asked when the inquiry will be completed.

In practice, enquire and enquiry are more common in British English, and inquire and inquiry are more common in US English, for both informal questions and formal investigations.

So, it’s up to you which spelling you use, though if you’re writing for a particular publication, it’s worth asking about their house style. Sticking with inquire is probably best if you’re at all unsure, and whichever you pick, be consistent!