A gamut refers to a subset of colors within a colorspace. In Photoshop, you may have seen gamut warnings. This happens when the color chosen is out of the gamut of the current simulation profile.
A rendering intent is the method used by a CMM (Color Matching Method) engine to deal with colors from a source profile that are out of gamut. Say you have a picture of a colorful flower with bright reds and greens, and you’ve tagged your image as Adobe RGB, and you’re having the image printed using the GRACoL2006 CRPC. Most of the image is probably reproducible in GRACoL2006, but the brighter saturated reds are probably out of gamut. The rendering intent algorithm is responsible for dealing with those out of gamut colors. You have 4 choices.
Choose Your Destiny!

Absolute Colorimetric

The absolute colorimetric intent clips colors that are outside the destination’s gamut to the parameter and colors that are inside the destination’s gamut are preserved. This rendering intent is the most accurate reproduction of the 4 rendering intents as it also maintains the white point of the source. This is commonly used in contract color proofing workflows as it simulates the paper color (White Point) of the source.

Relative Colorimetric

The relative colorimetric intent is like the absolute colorimetric intent in that it clips colors outside the destination’s gamut to the parameter, but the white point is inherited from the destination. Colors that are inside the destination’s gamut are preserved, but their white point is inherited from the destination. The relative colorimetric intent is the 2nd most accurate of the 4, as the colors that are already inside the destination’s gamut do change a little, depending on how different the source and destination’s white points are.

Perceptual

The perceptual intent moves all colors outside the destination’s gamut inward but maintains the distance between colors while doing so. Also, it does this without clipping color to the destination’s gamut. Depending on the content and size difference between the source and destination profile, you could see a large change after the color conversion. This is because all colors are moved. If the source and destination profiles are similar in size, you may see a natural conversion, with no color clipping. If the source is very large, like Adobe RGB, and most of the content is using colors outside the destination’s gamut, you’ll see a large color shift after conversion.

Saturation

The saturation intent tries to preserve as much saturation as possible. Each CMM handles the saturation intent differently and I don’t recommend using it as the results aren’t predictable.