What is the difference between a "request" and an "inquiry?"

A request is the act of asking someone to do something, while an inquiry is the act of asking for information.  Either can be expressed as a statement or as a question, though, which might seem confusing.  I will give you some examples.

A request:

Please turn up the heat in the house for me because I'm feeling chilly.

Will you please turn up the heat in the house for me because I'm feeling chilly?

Using the word "request" in a sentence:

I made a request at the hotel desk for a room on the first floor.

She will request a new roommate next term.

An inquiry:

Have you got this dress in a smaller size?

I would like to know if you have this dress in a smaller size.

Using the word "inquiry" in a sentence:

I need to make an inquiry about the availability of rooms next weekend. 

She will make an inquiry about which documents are going to be put into evidence.

These words are probably used interchangeably by some people, but they do have different meanings and uses.  It's always so much better to use language with precision!