If you ever use computer software like Photoshop, send images files to a printer, your website, other people, or take photographs with a digital camera, profiles will affect your work's tone and the colour of your work.
For example, if you buy Ultramarine Blue paint from two different manufacturers. The paint will probably be different in tone or hue due to differences in the mix of base pigments used. Similarly, manufacturers of computer devices, monitors, cameras, printers and printer inks use different base chemicals to produce their products. This causes differences in the way they behave and show colour. To ensure that a specific colour appears to be visually the same on different devices, colour-profiling was devised.
A Colour Profile maps the colour output of a device, be it a monitor or printer, to corresponding colours in the CIELAB colour space.
A colour space is an organisation or mapping of colours. It can be arbitrary with colours assigned to physical colour swatches and named as with the Pantone collection or mathematically structured as with the sRGB colour space.
The CIELAB space is a fixed colour space representing the four unique colours of human vision: red, green, blue, and yellow, plus brightness. (For a detailed explanation of the 4 unique colours in human vision: https://www.sciencedirect.com/topics/engineering/human-color-vision).
When you assign an image a colour profile, this helps software convert the colour values from one device to another.
For example, suppose your image is saved with the AdobeRGB colour space.
When you look at the image on a computer screen the computer will use this information to change the image colour values to that of your monitor's RGB colour space to represent the colours correctly.
If you then send your file to a printer, the software will then use the information to change the colour values to that of your printer.
Traditionally there were two main colour spaces.
RGB used on digital images creating the colours from Red, Green and Blue light.
CYMK used on printed media using the mixture of Cyan, Magenta, Yellow and Black inks.
However modern digital fine art printers use more inks than just Cyan, Magenta, Yellow and Black and are designed to mix their inks based on RGB colour space information. So it is better to save your file in the RGB colour space as it will then use all the colours your printer has on offer. These are the most common RGB Colour Profiles
sRGB IEC61966-2.1 (used as a web standard)
AdobeRGB 1998 (created to describe a larger space than the commonly used sRGB space. This is better to use should you wish to print your work)
Apple RGB
ColorMatch RGB
There are a large number of CMYK profiles too numerous to list.
CMYK space are usually smaller than RGB ones. It is usually better to keep your images in RGB for digital printing.