A review is a form of feedback to tell an author what is thought of their work, and offer opinions of the story and the skill of the writing. These are usually written and can be found in newspapers, magazines, and on booksleeves.
In fanfiction, fanfic writers often ask for reviews, but what they actually want, in most cases, is praise, and will not accept anything but the highest form of flattery. In these cases, constructive criticism is usually not wanted but often really needed. Unfortunately, this attitude has seeped into many of the popular websites, and not only do most of the readers leave excessively optimistic reviews for anything with a semblance of coherency and grammar, but the administrators will happily delete anything with a drop of negative in it as "abuse".
Reviews are often treated as a form of social standing within fanfiction websites (especially FanFiction.net). While there is some correlation to how widespread the exposure for a story is in the number of reviews, there isn't any correlation to quality of the work in question. A casual look at My Immortal (which, depending on who you ask, is either utterly horrendous or bad enough to become hilarious due to its ineptitude) will dispel that in a hurry.
For an example of what happens when an "author" does not learn how to deal with reviews, take a look at BigAl's review of "The Greek Seaman" by Jacqueline Howett. Read the review. Then read the comments. Laugh. Weep. Learn.