There are numerous color modes out there, but 4 of them (RGB, HSB, Lab, and CMYK) are most used in modern digital arts.

RGB: You probably heard this the most. This is the most widely used color mode for most artists. The name, RGB, came from the three primary colors of the additive colors-Red, Green, and Blue. Typically, RGB mode is 24 bits, and you can specify the brightness of each primary color in 256 different levels (0 being darkest and 255 the brightest). You have more than 16 million colors in this color mode. There is higher bit depth RGB modes available in many programs such as Photoshop, but many of their operations cannot be done in bit depth higher than 24-bit. Because of its wide acceptance, almost all the graphic file format supports RGB mode (except GIF). RGB is also the default operation mode for most programs, meaning all their operations and filters are done in RGB mode (even the image is in a different mode). Therefore RGB is also the fastest color mode. One drawback of this mode (same for the CMYK) is that this does not describe the color itself. It only tells what the display device should do and therefore it's device specific color system, and that's why if you see the same image on 10 different monitors, you'll see 10 slightly different images.

Three different 8-bit channels make up 24-bit RGB image

HSB (or HSV or HSL): Stand for Hue, Saturation, and Brightness (or Value or Luma). Hue (pure color) is a color range of 360 circle. Both saturation (purity) and brightness are measured by percentage from 0 to 100. This mode has about 3 million colors in its range.

Lab: Stand for Luminosity and two arbitrary color axes (a and ß). Unlike RGB and CMYK modes that describes how devices like monitors and printers generate colors (device specific), this color mode is based on the description of the actual color itself. So this is the only color mode that is independent of devices. Theoretically, this mode have the most widest color range. In fact, because of its wide range of colors, LAB is the intermediate color mode in most programs. For an example, if you chose to convert RGB file to CMYK, then photoshop converts the RGB to Lab first, and then convert it to CMYK.. Also Lab is as fast as RGB

CMYK: This is a color mode for printing. It stand for Cyan, Magenta, Yellow, and BlacK, and represent the amount of each ink required for the color. One drawback of CMYK mode is that it does not cover all the color range of the RGB mode. So when you convert RGB file, some colors (especially very bright colors) cannot be converted correctly. To help minimizing the artifact of the conversion from RGB to CMYK, many digital programs have "Gamut warning" functions which shows pixels with the colors out of CMYK range.

4 channels in CMYK mode

Hexadecimal: This is used in HTML. It's similar to RGB, but it's expressed in 16-base digits (1,2,3,4,5,6,7,8,9,0, A, B, C, D, E, F). This is very efficient expression since 8-bit colors can be described in 2 digits and entire 24-bit colors (16.7 million colors) in 6 digits (first 2 digits for Red, next 2 for Green, and next 2 for Blue). However, it's only used in HTML and very few other areas.

Grayscale: This is an 8-bit system with 256 different gray tones.

Following examples are some colors in different modes.
White: RGB- 255, 255, 255
HSB- 0, 0, 100
CMYK- 0, 0, 0, 0
Red: RGB- 255, 0, 0
HSB- 0, 100, 100
CMYK- 0, 91, 100, 0
Green: RGB- 0, 255, 0
HSB- 120, 100, 100
CMYK- 74, 0 , 96, 0
Blue: RGB- 0, 0, 255
HSB- 240, 100, 100
CMYK- 87, 85, 0, 0
Black: RGB- 0, 0, 0
HSB- 0, 0, 0
CMYK- 65, 53, 51, 100