There are so many different types of artists, styles, and methods. To provide good crits, you have to understand what the artist's goals are.
If they are aiming to be a realist painter, then crit them as such. If they are aiming to be a cartoonist, then crit them as a cartoonist. To crit someone who is trying to be a cartoonist using realist fundamentals doesn't make any sense.
That being said, learning the basic fundamentals of drawing, photography, or whatever your medium is a very good idea and will only help your more stylized work. If you draw a picture of a guy whose eyes are floating around on different levels and who has one arm shorter than the other, you can't just say "well it's my style." Style and a lack of fundamental drawing knowledge are two very different things. You can still understand those basics and apply them to the style you are using. Knowing proper anatomy, form, construction, lighting, color theory, and all of those other basic drawing fundamentals will only help your own work. Do not disregard those because you have chosen a style.
Most good anime artists, or other cartoonists have a strong foundation in basic drawing fundamentals and beyond. It's much easier to stylize the human form if you have a basic understanding of how it works.
Does that mean that you have to understand exactly how every muscle in the body works? No. Would it help you? Yes, it definitely wouldn't hurt you to know that much.
On the flip side, if someone is drawing a cartoony figure whose proportions, though stylized make sense for what it is, saying "the musculature curving on the calves is all wrong and you should have rendered it all perfectly realistically instead of using cel shading" is not a very good crit. You are disregarding what the purpose of the drawing is. They aren't trying to draw the perfect anatomically correct human form. They are doing a stylized drawing. Keeping in mind that there are more methods of drawing than the one you choose is very important.
If you do not have good knowledge of what the person is trying to achieve, then don't give detailed technical crits simply based on what you yourself are trying to achieve in your personal practice.
Which brings me to another point. "You drew a cat, but you should have drawn a dog because cats are stupid." This is a personal preference and NOT a constructive crit.
Erisian Pope gave me some great links to share:
Giving Crits: http://emptyeasel.com/2007/06/18/how-to-give-an-art-critique-constructive-criticism-for-artists/
Taking Crits: http://emptyeasel.com/2007/06/10/how-to-handle-artistic-criticism-learning-from-art-critics-artist-critiques/