Each browser has it's own ups & downs & I use all of them for various things...
Internet Explorer The main thing I do not like about Internet Explorer is not having the ability to disable the ActiveX (without getting pop-ups about it everywhere).
Ideally, ActiveX should be disabled by default & each site that wants to use it should be authorized by an admin or at least the user because ActiveX is very potentially dangerous. You can go to certain sites (porn, crack/hack, etc; etc.) & have programs installed without even asking you if you wanted it installed to begin with. Most of them are not that slick, mostly they tell you something like "click here if you want this download" & that's how they get you, however, I have come across sites that have been able to install software onto my computer without even a prompt about the installation. This is the main reason why only admins may use Internet Explorer on PC's I set up.
IE does however have some benefits. There are certain sites that actually require IE in order to function properly & not all of them are Microsoft sites like the Windows Update site. These sites are rare but they do exist. Most of the people that I know that use these types of sites are on home loan related type sites.
Opera The main thing about Opera is it loads fast. I use this for sites that have tons of stuff to load on 1 page. Currently I only use this for the GamingForums AdminCP because it takes a long time to load in Mozilla Firefox. I have a few different PC's & on some of them, the AdminCP loads decently well on them but not all of them. If I use Opera it loads fast no matter what.
Mozilla Firefox This is the browser I mainly use. It has an interface similar to Internet Explorer which newcomers tend to really like, it has a search bar built into it by default, as well as a totally customizable top bar.
Allot of times when you install search bars & stuff into IE, it can cause complications (normally when you have multiple bar add-ons installed), then you're pretty much stuck with the mistake because sometimes even when you uninstall the bar add-ons from IE, you still have problems & since you can't re-install IE, you're stuck with a messed up browser. With Mozilla, you could always re-install & solve the problem.
Mozilla also does not use ActiveX obviously (only IE does), so you don't have as much of a threat there, which is particularly useful when not-so computer literate friends or relatives come over & use your PC. Mozilla also has the ability to open pages in tabs which I find much better to use than having each new page in a new window.
All in all, Mozilla is my favorite but I do use the other browsers from time to time for specific reasons.