They say that AdobeRGB has more colors than sRGB color space.
What do they mean? Should they both have 16,777,216 colors (because they're both RGB color spaces)??
Adobe RGB also tends to be more red-ish, it's often used when printing stuff, sRGB is more common in the digital world with monitors and digital-media. They are different color-spaces, how many values they can represent it's something that is implementation-dependant, it's not the main difference between them.
In the color-world there are color-models and color-spaces, this 2 are both RGBs so they belong to the RGB color-model family but they are 2 different color-spaces.
The difference here is about how the arrange color values in the space and based on what, to really understand this you should study how a printer works from a color standpoint and how a display works.
It's like how the Mp3 compression works, you have an uncompressed audio source, you want to compress it or in other terms, to represent only the most significant Hz, so you need to make a choice AND you need to give an algorithm, a rule; in the Mp3 case the rule is simply what Hz the human ear can catch easier than others, you discard all the ininfluent or unheard Hz and you only keep the most important ones. In the RGB case you don't have audio but you got colors, you don't have Hz/waves but color-values/hues, and you always got a finite space, meaning that you need to make a choice because you can't store infinite values, so you basically make a choice based on what are the studies behind this 2 color spaces and in a nutshell the AdobeRGB is best for printing and sRGB is for TV, monitors and everything similar. To undestand how they works just read the papers, it's impossible to explain this in a short answer but there are many resources around the web.
Check out this article http://www.kenrockwell.com/tech/adobe-rgb.htm which basically says that they both occupy the same range, but due to software the Adobe RGB can be unpacked and then further expanded.
Adobe RGB theoretically can represent a wider range (gamut) of colors, however:
1.) Adobe RGB requires special software and painstaking workflow not to screw it up. Make one mistake anyplace and you get dull colors, or worse. You cannot use Adobe RGB on the internet or for email or conventional photo lab printing. If you do, the colors are duller
Further reading about the 2:
sRGB is a RGB color space proposed by HP and Microsoft because it approximates the color gamut of the most common computer display devices. Since sRGB serves as a "best guess" for how another person's monitor produces color, it has become the standard color space for displaying images on the internet. sRGB's color gamut encompasses just 35% of the visible colors specified by CIE (see section on color spaces). Although sRGB results in one of the narrowest gamuts of any working space, sRGB's gamut is still considered broad enough for most color applications.
Adobe RGB 1998 was designed (by Adobe Systems, Inc.) to encompass most of the colors achievable on CMYK printers, but by using only RGB primary colors on a device such as your computer display. The Adobe RGB 1998 space encompasses roughly 50% of the visible colors specified by CIE — improving upon sRGB's gamut primarily in cyan-greens.
So, it would appear that the colours available to RGB is much greater than what is offered by either sRGB or Adobe RGB and that Adobe applies a compression technique as well as choosing to select a different range of colours (compared to sRGB).