µTorrent is a BitTorrent client. A 'client' in this case is a computer program that follows the rules of a protocol. For example, HTTP (HyperText Transfer Protocol) is the protocol used to download web pages and other content - like this page - and your HTTP client (or browser) is the program you use to get those web pages. Some popular browsers include Microsoft Internet Explorer, Mozilla Firefox, Safari, and Opera. To an extent, they all work the same way because they follow the same set of rules.

Just as there are multiple HTTP clients, there are multiple BitTorrent clients, and µTorrent is one such BT client.

How do I download files using BitTorrent?
Just like you need a URL like 'www.google.com' to go to a web site and download content, you need a 'torrent file' (often abbreviated as '.torrent' or just 'torrent') to download content available using BitTorrent. Most of the time, you will download this file from a website, though you could also get the torrent from a friend or even a CD. Many websites offer torrents as one method of downloading files available through that website - for example, OpenOffice.org, a free office suite, can be downloaded using BitTorrent. Other sites, like legaltorrents.com, offer torrents of all kinds of things - these sites are just repositories of torrents and usually don't actually create any of the content available. They're known as indexes or trackers - there is a subtle difference between the two. (The Wikipedia article on BitTorrent trackers explains the difference.)

Once you've obtained the torrent file from wherever, you simply need to import it into your BitTorrent client (hopefully µTorrent!). There are several ways of doing this in µTorrent:

Click File then Add Torrent in µTorrent (or press CTRL+O) and locate the torrent file.
Double-click the torrent file. (Only works if you've associated .torrent files with µTorrent - µTorrent asks you if it should do this the first time you run it. If you clicked 'No', you can do this by going to Options, then Preferences in µTorrent, then clicking Associate with .torrent files under Windows Integration.)
(advanced) Click File then Add Torrent from URL in µTorrent (or press CTRL+U), and enter a URL from which the .torrent file can be obtained.
If you're using another BT client, consult its documentation for instructions.

Two common mistakes made by BitTorrent newbies
The two most common mistakes made by BitTorrent newbies are not forwarding ports and not seeding torrents.
If you don't forward your ports and don't allow µTorrent full access on your software firewall (if you have one), you will not get the most out of BitTorrent (this page explains why). You can get help on port-forwarding at portforward.com, and some (slightly technical) information is available at Brian's BitTorrent FAQ and Guide (note that µTorrent does not use 6881-6889, nor should those ports be used).
Seeding is where you leave your BitTorrent client open after you've finished your download to help distribute it (you distribute the file while downloading, but it's even more helpful if you continue to distribute the full file even after you have finished downloading). Chances are that most of the data you got was from seeds, so help give back to the community! It doesn't require much - µTorrent will continue seeding until the torrent is removed (right click the torrent, then hit Remove). However, it is important to be aware that while seeding, µTorrent is uploading using your internet connection, and if your internet service provider limits the amount you may upload in a given period, you may wish to monitor how long you seed torrents for.

But I don't want to download software, I want to download other files!
Most people use search engines, such as Google, and add the term "torrent" to their search. BitTorrent can be used to deliver any type of file. BitTorrent is purely a content distribution method and (just like a web browser) does not incorporate any technology to differentiate between content that is legitimately shared and content that is pirated. As a result, BitTorrent is not designed to protect you if you're doing something illegal. If you're using BitTorrent, anybody can see your IP address. In many countries, an IP address, time, and evidence that you're doing something illegal is all that is needed to prosecute you.

How do I know that someone isn't sending out viruses on BitTorrent?
In short, you don't. You should treat something downloaded with BitTorrent just like any file downloaded from the internet - that is, if you don't trust the source of the file, then you should use caution when opening it. Running a virus scan is usually a good idea. However, BitTorrent does have protection to ensure that someone uploading parts of the file you're downloading (ie. someone who is not the original source of the torrent file) is not uploading malicious data to your computer.

Warning: the following paragraph is relatively technical!
This is done using the torrent file. Not only does this file contain the information needed to find people with parts of the file and start downloading off them (the tracker URL), it also includes hashes of each piece in the torrent (remember how BitTorrent splits each file up into many smaller pieces?). A hash is, put simply, a small amount of data that uniquely represents another piece of data and is calculated using a one-way function - meaning that it is easy to find the hash of a file, but almost impossible to find a piece of data that would generate that hash. So technically someone could be sending you malicious content instead of a piece of the file - but when you receive it, BitTorrent would hash the malicious content, discover that it did not match the real hash, and discard it. (Still want to know more? Read up on cryptographic hash functions or SHA-1, the hashing algorithm used by µTorrent.)

So, while BitTorrent makes sure you get what you asked for when you downloaded the torrent file, it can't guarantee that the torrent file wasn't malicious in the first place - which is where anti-virus software comes into play.