The traditional distinction between enquire and inquire is that:
Enquire is to be used for general senses of ask while -
Inquire is reserved for uses meaning make a formal investigation.
According to the Macquarie Dictionary, some organizations such as newspapers tend to prefer the in- form, but a distinction should be made between, for example, an official inquiry and an informal enquiry.
Enquiry is used when asking a question or seeking information.
For example, a journalist will phone a PR professional and make an enquiry about a topic.
Inquiry is for official investigations.
For example, the senate will hold an inquiry into fuel prices.
In practice, however, enquire (and enquiry) is more common in British English while inquire (and inquiry) is more common in US English. As a matter of fact, it is only in British English that any attention is paid to the distinction. In US and Australian English, inquiry has, for all practical purposes, taken over. Otherwise there is little discernible distinction in the way the words are used.