When to use  
One reason for is to insert multiple spaces in a document.
In HTML, multiple whitespace characters are collapsed into one space. This includes tabs and newlines.
IF you wanted to display the following:
three spaces.
You could insert 3 entities instead of using spaces like so:
three spaces.
Edit: It's worth mentioning that is more of a historical artifact than anything else. Just about every use for it that is mentioned in the answers to this question has a better alternative means to accomplish that goal. However, is still with us, and these are some of the things people have used it for.
See also: http://www.sightspecific.com/~mosh/www_faq/nbsp.html
I don't know if this answers your question or not and certainly this answer is not of the caliber already provided by others, but the beauty of a discussion thread or Q&A site is the diversity of experience that might be found in it. So, on that note, I'll share with you what I've used nbsp; for. (To be perfectly honest, 24 hours ago, nbsp; was something I had never even heard of.)
Here's how I used nbsp;. I was posting something using markdown language and I had a very simple two-item bulleted list. For the life of me I could not get the spacing before this list and after to look symmetrical. So, I did a web search and somehow ended up taking a look at this thread.
Before using nbsp; the paragraph that followed bullet point #2 collapsed the spacing between the bulleted point and the text, making it look as if the paragraph had something to do with bullet #2, specifically (which was not the case). I tried a lot of different things that I can't even remember now, but the one thing that ultimately worked was insertion of nbsp;.
Since then, I've been seeing all sorts of posts that indicate some controversy over its use, but for non-coders who need to wrangle out of an unsightly/misleading formatting issue, nbsp; is a very quick and useful fix.
(it should have a semi-colon on the end) is an entity for a non-breaking space.
Use it between two words that should not have a line break inserted between them by word wrapping.
There is a good explanation about when this is appropriate grammar on the English StackExchange.
It is sometimes abused to create horizontal space between content in web pages (since it will not collapse like multiple regular spaces). Padding and margins should usually be used instead of this hack.