Google Chrome. It's fast, easy, and has the "Incognito Window" feature (although Opera has a private tab). [I love Opera's features, but my laptop won't let me download it. :c]
Wait, won't let you download it..? Why not? (If I may ask)
And as noted above, most browsers have some form of private browsing... Firefox only does it by session though, Chrome does it on a window basis, and Opera is by the tab. I find it hilarious that there is a shortcut to close all private tabs but leave the others open in Opera though. ([Ctrl] + [Shift] + [Q]) :D
Btw I saw interesting test somewhere. They were testing start up speed, loading speed of websites and battery consumption while those browsers are running. Intersting thing was that all those tests won newest IE 10.
IE did huge progress in user friendliness, support of standards and performance. Microsoft really started to work hard with it. Question is if it is not too late, because bad name which they made is just too strong.
I can say from experience that IE10, when first loaded is slower than Chrome. (But faster than Firefox, and about the same speed as Opera)
Btw with safety and usage of modern technologies are all major browsers on nearly the same level. What counts for me atm is implementation of HTML5. Chrome got the best, but that is because of fact, that main designer of HTML5 is employee of Google (or something like that). Anyway it is not standard yet so I expect Firefox will reach the same level as soon as it will be out as standard (difference is already not that big).
This is untrue...
HTML5 came from Opera. (You only need to read the first line)
And the best desktop implementation is actually from Firefox currently. Best overall is from Opera Mobile, but not their desktop browser yet. I am not sure about
"Next" though, it probably implements the full API, I haven't fully checked. (You can try Opera Mobile via their official Android version emulator
here)
You can check your browser's API compatibility by checking out Opera's HTML5 and CSS3 API showcase
here.