Last time we looked at the difference between RGB and CMYK colors. To recap: The RGB colorspace is smaller than our eyes can perceive, and CMYK is usually even smaller:
As you can see to the left: Our perception (the upside down parabolic field) is larger than what RGB can display, and CMYK is even smaller (but may be able to display some colors RGB can't, which depends very much on the printer used).
To the right you see how Green looks in CMYK and in RGB. Although they should be equal in theory, you get a big difference in color. Just absorbing colors can't display the really strongly shining green RGB can. CMYK can also not reproduce fine nuances in highly saturated colors. For instance the Full RGB-Green becomes a muddy green following conversion (the two boxes in the image).
Rendering Intent: Perceptual
This rendering intent keeps the relative distances between the colors the same, allowing for a picture that looks perceptually the same. This keeps the overall look of an image the same, and is usually the best for photos.
The original colors are moved inside the color space leaving the relative space between them intact (here the original colors are the red rings, the blue ones are the converted ones. The black ring is the white point and the reference point for all conversions.
Rendering Intent: Relative Colorimetric
This rendering intent to keep the colors as relative to their original position as possible:
Colors inside of the space the printer can print are left where they are (green ring), the others are moved to the next color just inside of what the printer is capable of. This means that color nuances can be lost as, like in the example, two colors fall onto one color in the result. Depending on the image this can either be ignored (like the black noise in a jpeg that should be black) or take from the image (different greens on a field of grass blur into just one ugly blob).
Rendering Intent: Saturation
Want to print a logo or pie charts? Then you might want fully saturated colors, and this is the rendering intent to go for. This will try to keep fully saturated colors as pure and saturated as possible. This might even change the hues of the colors.
As you can guess from the image above, this should never be used for photographs - unless you want to use it for artistic purposes.
Rendering Intent: Absolute Colorimetric
And now I have to admit, that I have been lying to you in the post about Relative Colorimetric: The white point (the black ring) and the black point (not in the sample space I quickly made using Inkscape) are not usually in the same point in different color spaces. Relative Colorimetric compensates for that as well, moving those points along and moving all colors according to the new white and black points - that means even those that are already inside the colorspace!
Absolute Colorimetric ignores these corrections for the new white and black point, and absolutely leave the colors that fall neatly into the new color space where they are ... This can be desirable if you want to render special "signature colors" in an image ... a rendering of a company-logo for example.
This concludes this small excursion into color spaces and rendering intents. This was something I had to do some reading up on, but hope that now that I have written all those down and made these images, that I am now better at remembering what to use when.
TL;DR
If this is too long, and you don't want to become an expert: Perceptual is the standard rendering intent to use and should give the best results.