Arial
Tahoma
Verdana
Calibri

The mystery behind the prevalence and your exposure to each of these font faces can be answered in two words: Microsoft Windows.

Each can also be associated with particular versions and sections of the MS Windows user interface, and thus time periods as well. With these four fonts, each being sans serif, the raster representations of them were carefully measured by Microsoft, to provide users with highly legible, practical type faces.

Arial: the oldest of the standardized MS font package shipped with Windows, and used heavily on printed marketing materials and as logo type since Windows 95 and maybe earlier. The flagship font for Microsoft for many years, contributing greatly to its prevalence.

Tahoma: with Windows 2000, tahoma was the default UI font, with lucida console being used by notepad.exe (although notepad and the UI were both configurable), and many of the font fields where users entered data. This held its position until roughly 2007. People used it because they new it would be reliably present.

Verdana: 2nd only to helvetica among many digital graphic designers, and the helvetica stand-in on windows. In fact, because of the absence of helvetica by default on windows (you had to download it ,or purchase it, or 3rd party software that came with it), as opposed to macintosh which provided helvetica in it's font set out-of-the box, and yet the market share of microsoft dominating over apple, more websites balanced their design to render well with verdana over helvetica. More often than not, you'll see CSS styles applied in the following order: helvetica, verdana, arial, sans serif.

Calibri: the reasons you dislike this font are two. reason one, the much-hated windows vista is where this font made its debut as the interface default on windows, so there's probably some negative psychological aftertaste hanging in the air, what with all the pain vista inflicted. reason two: the abominable CLEARTYPE sub-pixel font renderer which first appeared with Internet Explorer 8 on windows XP, but became pervasive and omni-present with windows vista, and truly made vista look like SHIT. Screenshots of text on vista were forever contaminated by cleartype. the groupthink of focus group testing produced nigh-infallible statistics absolutely proving to so many very important decision making people that discriminating users who know, always preferred cleartype. I suspect that the testing was influenced by the display monitors tested on. cleartype rendering was improved by the time Windows 7 was released, and monitors were better by then too, but no one cared, and Microsoft's market momentum was absolutely destroyed by then, so it didn't (and doesn't) matter anymore.

Which leaves us with the two serif fonts you made mention of...

Georgia: again, a Microsoft font. I think this is gaining popularity among Windows Phone users, but I'm not entirely certain of this.

That leads us to the inevitable...

Times New Roman: the un-killable highlander of fonts. There's a reason why it's used everywhere as the de-facto, ultra-generic fall-back default, especially in web browsers. Everyone can use it royalty free, and it's the one thing, even blood-thirsty competitors will reliably provide as common ground across platforms. They won't get sued for it. Strangely, even though it may be used royalty free, it's not truly a public domain font. (...owned by News Corp? weird.)