These days, it takes more than some nice content and a natty design to get one up on your competitors' websites. Slick interactions, speed and a great user experience are the name of the game. jQuery helps us achieve all these things.
If HTML and CSS are the meat and potatoes of web development, Javascript is the garnish. HTML is used to describe the content of a webpage and CSS is used to describe how that content looks. Javascript is a programming language that web browsers understand that can be used to manipulate HTML and CSS based on input provided by the person browsing the webpage.
jQuery is a library of Javascript functions that can be called upon and combined to perform complex actions in the browser, based on defined behaviours that the user can trigger. These actions can be performed without having to reload the webpage from the server, meaning they occur much faster than they would otherwise.
jQuery (and the underlying Javascript) can be used to manipulate a webpage's HTML and CSS in all manner of ways. These are most often used to improve usability and the user experience, but can also be used for purely aesthetic effect. For example, on this website the 'parallax' effect created by the wallpaper moving at a different speed to the speed at which you scroll is created using jQuery to animate the background based on the speed at which you operate the mouse scroll wheel.