jQuery is a JavaScript framework designed to allow developers to literally write less, do more, the 3 different flavours you are asking about, do very different things.

First up jQuery is the core library which contains the main functionality of the framework, so if you want to make an element fade in, you would include jQuery on your page, and then call the fadeIn() function on one of your elements.

jQuery UI was created with user-interface elements and interactions in mind that generally require a lot of code to manipulate otherwise. So the library contains things like the ability to drag and drop elements, and get useful information from those interactions. But at the same time there is a focus on the look and feel, therefore there are all sorts of themes available to style these powerful UI elements in this library.

jQuery mobile is a powerful framework built on top of jQuery and designed specifically for mobile platforms - after all the functionality of things like drag and drop needs to be rethought for touch screens, as does core functionality such as hovering which simply doesn't exist on mobile.

All together the frameworks can be used to enhance the usability of a site, and really save developers a lot of time. To learn more about the framework visit Learning jQuery - a great resource for getting started.

Plugins allow web developers to take the solution to their problem and abstract it into a single functional and independent block (usually in a separate file). They can then share that with everyone else so that they can do the same. So if someone builds a gallery feature for a site they made, making a plugin would allow them to easily reuse the code, and help others facing the same problem.