If we have an RGB image, most browsers and, in fact, monitors only support sRGB space. I am trying to understand something important. Does the monitor/web then convert each of the pixels in the image to sRGB and then display it? Meaning we are actually seeing the sRGB version of the image.
Also, if that is the case, which formula can we use to do the conversion, and if we did the conversion ourself, I assume we would get an image that 'looks' exactly the same as the original?
The pixel values in an image file are just numbers. If the image is "in" a space larger than sRGB (such as ProPhoto), the pixel values will only be "converted" to sRGB if you have color management enabled, OR you perform the conversion yourself.
A browser or device will only convert tagged images of a non-sRGB colorspace TO sRGB IF there is a color management engine.
With no color management, and a standard sRGB monitor, all images will displays "as if" they were sRGB, regardless of their colorspace. I.e. they may display incorrectly.
Even with color management, if the image is untagged, it will be displayed as whatever default (usually sRGB) the system is set to use.
As for formulas: the conversion is known generally as "gamut mapping" — you are literally mapping the chromaticity components from one space to another. There are multiple techniques and methods which I will discuss below with links.
If you want to do you own colorspace conversions, take a look at Bruce Lindbloom's site. If you want a color management engine you can play around with, check out Argyll, and here is a list of open source tools.
Because there are other answers here with some "alternate" (spurious) information, I'm editing my answer here to add and help clarify.
And to be clear (in reference to another answer): RGB is a color MODEL, sRGB is a color SPACE, and a parsable description like an ICC profile of sRGB is a color PROFILE.
Second, sRGB is the STANDARD for web content, and RGB values with no profile displayed in a web browser are nominally assumed to be sRGB, and thus interpreted as sRGB in most browsers. HOWEVER, if the user has wide gamut (non-sRGB monitors) and no color management, then a non-color managed browser is typically displaying at the display's colorspace which can have unexpected results.
RGB is an additive COLOR MODEL. It is so named as it is a tristimulus model that uses three narrow band light "colors" (red green and blue) which are chosen to stimulate each of the eye's cone types as independently as possible.
sRGB is a colorSPACE. A color space is a subset of a color model, but adding in specifics such as the chromaticities of the primary colors, the white point, and the tone response curve (aka TRC or gamma).
sRGB-IEC61966-2.1.icc is an ICC color PROFILE of the sRGB colorspace, used to inform color management software to the specifics such that appropriate conversion can take place.
Color Management is a system that uses information about device profiles and working color space to handle the conversion for output, viewing, soft proofing on a monitor, etc. See This Crash Course on CM
LUT or LookUp Table is another file type that can be used to convert images or apply color "looks".
Gamut mapping is the technique to convert, or map, the color coordinates of one color space to the coordinates of a different color space. The method for doing this depends on a number of factors, and the desired result or rendering intent.
Rendering intent means the approach used, as in, should it be perceptual? Absolute colorimetric? Relative with black point compensation? Saturation? See this discussion of mapping and Argyll.
Colorspace transforms and conversions are non-trivial. It's not as if you can just stick image data through a nominal formula and have a good result.
A reading through the several links above will help, and I'd also like to suggest Elle Stone's site. particularly regarding profiles.