It was never the intention of copyright law to prevent all uses of a copyrighted item. In the academic world people will often quote or cite other copyrighted works in their own work and this use is protected under the principles of fair use.
Fair use as a principle is a very gray area. There are no quantitative measurements as to how much of a work you're allowed to use. In a legal situation it will come down to the judges opinion. If faced with a fair use situation you should consult your lawyer, as they should have a good idea of what you can defend in court as fair use.
There are four main considerations to apply when dealing with fair use.
If the purpose of the use is for an educational or nonprofit purpose you're on the right track. Using a small portion of a work for non-commercial purposes is permitting under fair use as long as it fulfills the other considerations. If the item is being used for commercial purposes and is generating money then it is not protected under fair use.
The nature of the work hinges on fact or fiction. If the work is of a factual nature, such as a newspaper article, using a portion of it is more likely to be allowed under fair use than if the work was of a fictional nature, such as a novel or poem.
The size of the use is an often misunderstood issue. Many think that there is magical percentage of a work that they are allowed to use. This is not at all true. This is just one of many factors and the decision will ultimately come down to the judge. So don't bank on a percentage to protect you from being sued. Additionally many people believe that you can borrow someone's work, change it substantially by a certain percentage, and publish it yourself. This is not true, in fact you've just created a derivative work which is against copyright law.
The final, and perhaps most important consideration, involves the impact of your use on the potential market. If your action in any way decreases the amount of revenue collected by the author for their work it will not be protected by fair use. A teacher who copies a book so that her students do not have to buy it would be breaking copyright law, even though she is using the book for educational purposes and is not profiting from it, because her actions lessen the revenue for the author.
There are a few exceptions to fair use law as well. If you're using the work for scholarly criticism or for parody it is allowed. This is why Weird Al stays in business and why movies like Space Balls or Scary Movie can be made.
As you can see fair use is very ambiguous. If, after applying the above principles to your site or other work, you are unsure on whether or not your use of other's work would be protected, you should contact a lawyer.