First, the nutshell answer: structured data is everything a search engine would want to know about a webpage, all organized by agreed standards and written into the page code for easy access by all search engines.
Now, starting over with a little more detail (and ending with Why You Care):
Webpages are created using HTML—which is a markup language, not a programming language. In other words, you can't use HTML to write programs that compute things; by itself, HTML can only organize textual information and display it for you.
HTML organizes its information by “marking it up,” by adding tags and brackets around pieces of information. Some of this is just cosmetic, but some of it is quite functional; for instance, an H1 tag is a very clear sign to search engines that “this thing is a headline, treat it accordingly.”
Our last piece here: some of the stuff written in a page's HTML isn't visible on the front end. A lot of this stuff is called metadata (because it's data about the rest of the data), and this page metadata is where search engines get a lot of their clues about how to interpret a page's content.
Which brings us to our more specific definition of Structured Data: it's a specific part of each webpage's HTML metadata specifically intended for use by search engines.
So now what?
Why You (Might) Care
Nutshell Answer: Proper use of structured data is one criterion for “special treatment” within Google's search results.
More broadly, implementing structured data on your pages is just “good SEO form” and will make it easier for search engines to serve your content to others. But it's important to clarify that it's up to each search engine what they do, exactly, with the structured data; if we're being realistic and ignoring everyone but Google, it's enough that Google puts weight in this.
More specifically: Google uses structured data (or parts of it) to evaluate who qualifies to have rich results displayed on Google's search-result pages. The idea here is that (A) Google wants to be able to show rich results directly on their results page, but (B) they need more information from content publishers to be able to identify those rich results accurately and reliably.
We'd definitely recommend Search Engine Journal's beginners' guide here if you want to learn more and see how this applies to you.
You could also skip straight to Google's Webmaster Blog for posts like this one, which explains how to include licensing information with your photos for parsing by Google Images (and potential use in Google results thereafter).