Question: I have a number of projects that could benefit from JavaScript. I've seen a lot of "Oh, you can use JavaScript for that!", but little detail about how and where that's done. Does the script live on a server somewhere or is it just embeded with a cut-and-paste via the "edit HTML source" button?

Looking for some guidance on the best place within the sites/farms from which to run scripting.

Answer: 
Technically, the scripts can reside anywhere as long as the users have read access to them.

In practice, client side solutions usually have a short lifecycle, and you'll certainly want to place them in a location that you can easily update. A good option is the Style Library within a site collection, which has the added benefit of caching the files. If you have internet access, you could also consider using a CDN to call popular files (like jQuery.js).

On the pages themselves, you can add the scripts to either the master page or via a Web Part, depending on the scope of your customization. Here are some explanations from my blog if you use Web Parts:

<http://blog.pathtosharepoint.com/2010/10/27/about-scripts-web-parts-and-urban-myths/>

[Update] I just came across this fresh article:

<http://techtrainingnotes.blogspot.com/2012/05/adding-javascript-and-css-to-sharepoint.html>

Note that the above links only show some basics. There are several more ways to include scripts, especially if you work with Master pages.
There are several ways to add JavaScript to SharePoint. You could add directly or reference it into the masterpage, page layout, code behind etc.. The script lives always on a server, because it serves it. 

Keep in mind that JavaScript is different language to learn and can be very powerful. Therefor I recommend learning it:

<http://www.codecademy.com> (Nice website where you can learn the fundamentals)
<https://developer.mozilla.org/en/JavaScript/Guide> (Very comprehensive information)

Besides learning the language it's recommended to place JavaScript into files, so you can reuse and manage your code. The **Style Library** is a good place for the \*.js files, also such as for the style sheets (css). Besides learning JavaScript I would also recommend to have a look at jQuery.