Well, simply speaking, the biggest bonus I see most people cite about Firefox is that it's generally more secure. It's open-source, meaning that anyone can look at the code-- and while that sounds like a bad idea at first for a browser, it means that the security holes get fixed much faster due to having an entire community working on it.
Microsoft, on the other hand, doesn't update IE nearly as much as Firefox gets updated. When Firefox has a problem, especially with security, an update is released pretty quickly, but IE takes much, much longer for any updates to get out. Or at least, that's how it used to be. I haven't used it since IE7.
There are also tons of great addons for Firefox, many of which increase the browser's security even more.
I can only tell you about Firefox since it's what I use, but I hear a lot of people like Chrome too.
Being a user of various browsers (Internet Explorer*, Firefox** and Chrome***), i can say that using either an alternative browser to Internet Explorer is still recommended.
IE is on version 9 at the moment, and with Windows 8 coming in the Autumn, there is the impending release of IE10. Microsoft claim it is more secure, but i'm not convinced and neither are other people. As Tonch mentioned, browsers like Firefox, Chrome, Safari, Opera etc are updated more frequently. And as such, some are on version 18.
Plus, by using an alternative browser, you get the joys of add-ons. Such as colours for tabs, social network add-ons, news add-ons. You name it, there is most likely an add-on for it.
I mean, there are loads of browsers out there for taking a look at. Me personally, i recommend leaving the dreads of Internet Explorer behind and jump over to something more stable and safe
* Only used to install an alternative browser, and to sort out my Mum's account
** My main browser
*** Alternative browser, also used to sort out Dad's account.
The only reason you need to know is the lack of web standards in IE. For whatever reason, Microsoft ignores the w3c and just doesn't implement web standards. Something can look one way in every other browser because you're just using standard code from w3c but it'll either be broken or just not there in IE. It's stupid.
I'm still waiting for a browser that doesn't have periodic memory leaks.
The best thing to do is try them all out and see which one best fits your style. If you're going to use Firefox, but she to download some nifty addons. Firefox is more suited for "power users", but it's certainly simple enough that anyone else can use it as well.
At any rate, Chrome, Firefox, IE, etc. are starting to look and feel very similar to each other (except that IE sucks) to the average user. So I wouldn't worry about "which web browser is the best" too much.
IE 9 actually isn't a bad browser, it's quite standards-compliant, fairly fast, and generally pretty decent.
I mainly like Firefox 'cause it's got the lowest memory footprint of all of 'em (With extensions that stops being true, but the core browser is very lightweight) and because it's easily the most customizable.
Yeah, once you put addons into Firefox the thing is a monster, which is kind of a problem. They tried to mitigate that with the whole "plugin container" thing, but sometimes that makes the memory footprint WORSE.
I just can't really disable any of my addons, I use them too often. And half of them are security-related.