The jQuery homepage says:
jQuery is a fast and concise JavaScript Library that simplifies HTML document traversing, event handling, animating, and Ajax interactions for rapid web development. jQuery is designed to change the way that you write JavaScript.
... and that's farily concise.
Using jQuery allows for very powerful JavaScript to be constructed quickly and with minimul fuss. jQuery also standardizes the incompatible JavaScript event and external CSS models. This means that cross-browser code can be written quickly and simply.
jQuery also makes implementing transition effects, something which can be laborious in JavaScript, the matter of one or two lines of code.
jQuery links to HTML in exactly the same way as any other external JavaScript file. Assuming the jquery.js file has been placed in the same folder as the HTML page you want to link it to, this code will work:
<script type="text/javascript" src="jquery.js"></script>
It would be best to place this code in the <head> of your page.