Sorry for the title, I didn't mean to insult anyone, just wanted to get your attention. Now that I have it, let's talk about color profiles and color management.
Without going too deep into color theory or technical specifications, lets just me just say that color profiles are a tool meant to help content creators make sure their audience sees everything the same way they do. This blog will focus on digital cameras and digital images, but the same principles may apply to analog media and other things.
When a digital camera records an image, it embeds the image file with a color profile information based on the camera's color space settings. When this image is then reproduced, that color profile is also read. If possible, the source colors are matched with the color capabilities of the reproduction media with the help of the color profile information. This is called color management. Color management makes it possible to view the colors the way they were recorded no matter what medium you are watching them on.
Different color profiles have different uses. Your camera might record 14-bit image with wide color space so that all the possible color information is saved. When you then edit the image and convert it into compressed JPEG image with 8 bits per color channel, some information is lost. Wide gamut profile is no longer needed, smaller profile is enough. If image is meant for printing, the display profile must be converted to a print profile. If the management is done correctly, the image will look the same everywhere.
While editing your images, you should save all the information available as long as possible and do the conversion as the very last step. In practical terms, use color profiles with wide color spaces while handling raw files (or other files that support more than 8 bit colors) and export the files into JPEG images using more narrow color space profiles make sure the colors are displayed correctly everywhere.
Same in even more practical terms: Check your image editing software's color settings before starting to work with the images. Use Prophoto RGB profile while editing raw files across Lightroom and Photoshop. When you are done, export (use export function instead of save function) files into JPEG images and choose either embedded sRGB or Adobe RGB color profile. sRGB is widely supported color profile and should make sure your image is seen as you meant it everywhere. Adobe RGB's purpose is to match print profiles with CMYK spaces closely for more faithful color reproduction in printing.