Anorton has already explained how to produce a simple line break in Markdown. I'd like to take this opportunity to suggest that you probably shouldn't use it.
Put simply, there are two conventional ways to separate paragraphs in English typography:
You can put a blank line (or at least some vertical whitespace — many style guides recommend somewhere between half a line and a full line) between the paragraphs, as in this example.
When you do that, you do not need to indent the paragraphs.
Alternatively, you can leave out (most of) the whitespace between the paragraphs, and just have the second paragraph start immediately on the next line.
When you do that, you should indent the first line of the second (and subsequent) paragraph, to provide a clear visual indication of the paragraph break.
This style works best with narrow columns, where most paragraphs will be long enough to span multiple lines.
Single-line paragraphs can look kind of funny with this style.
The Stack Exchange style sheet is designed with the first (blank line) paragraph style in mind. While you can sort of fake the indentation style (as demonstrated above), it requires hacky and convoluted markup, and frankly just doesn't look very good here.
If the SE designers had wanted, they could've written the style sheet here so that normal paragraps (separated by a blank line in the Markdown source code) would be rendered using the indentation style, with an appropriate amount of spacing and indentation to make them pretty and readable. But they chose not to use that style, and IMO, we really should not try to fight that choice.
In any case, whichever paragraph style you prefer, you should pick one of them and use it consistently — using both a blank line and indentation to separate paragraphs looks silly, and using neither makes it hard to see at a glance where the paragraph breaks are.
So, what are the legitimate uses for line breaks in Markdown, then? IMO, there aren't many, but they do occasionally come in handy:
Markdown doesn't really have proper support for image captions, but you can fake them pretty well by putting the image on one line, followed by a simple line break and the caption. (I also like to wrap the caption in
</sup> tags, to make it look like it's set in small type. This is a total abuse of HTML markup, but it works reasonably well.)
Sometimes, you may wish to format a list in a way that Markdown doesn't properly support (e.g. using letters instead of numbers in an ordered list). Again, this can be reasonably well faked using line breaks (although you can't get proper list indentation that way).
Perhaps the only non-hacky use of line breaks would be to format text that is naturally structured in lines rather than in paragraphs, such as line-based computer program output, or maybe poetry. (Note that, for program output, code blocks may sometimes be preferable.)