Paper shredders are used for destroying documents containing sensitive information, so in general you'll want to get the nicest one you can afford, unless your shredding needs are very simple. For the average schmo (that's you), the biggest worry is identity theft. Credit card bills, bank statements, tax information, and other such documents contain enough information for a third party to assume your identity; racking up debt, ruining your credit, emptying your bank account, or making small contributions to wealthy individuals in Nigeria who need to move large sums of money into the United States. Businesses have bigger concerns. Office memos, reports, schematics, legal documents, and many, many other types of inter-office correspondence can be embarrassing or damaging for even honest and legally run businesses if they were found by the wrong people. Competing companies would be able to gain an unfair advantage if certain trade secrets and business practices could be found. For these reasons, paper shredders have become increasingly popular in the home and office for ensuring that confidential information remains confidential.
There are two major kinds of paper shredder generally available today for home and office use, regular and cross-cut. Regular paper shredders use a row of circular blades powered by an electric motor to simultaneously feed and slice paper into narrow strips to destroy a document. Cross-cut shredders have an additional mechanism to cut these strips a second direction, resulting in a confetti-like collection of tiny paper rectangles rather than long strips. Although more expensive, cross-cutting is generally considered to be a more secure form of document destruction. Additionally, cross-cut pieces take up less room than regular cut, because regular cut strips fold over, leaving large air gaps, and must therefore be emptied more often. Most paper shredders come with a catch bin, but some are designed to fit over a garbage can.
Regular cut paper shredders are less secure because the long strips can, with a reasonable degree of patience and dedication, be reassembled back into the original document. Especially vulnerable are documents that have been shredded in the direction of the word flow, allowing complete sentences to be clearly read even without reassembly. Reassembly of cross-cut pieces requires powerful and very intelligent software (which to my knowledge has not yet been written) or at least severe obsessive-compulsive disorder and an unhealthy love for jigsaw puzzles.
In addition to these two basic options, paper shredders come in a variety of power options. The cheapest ones can only handle a few letter sized pieces of paper at a time. A stack of more than three or so pieces of regular 20 pound stock copier paper will cause the blades to jam or the motor to stall, and the paper will need to be removed and re-fed into the machine in smaller stacks. The cutters will usually not be damaged as long as this doesn't happen too often. Some paper shredders have a reverse option to back the paper out, others require the operator to physically pull the jammed paper out. It is very important to feed paper straight and square into the weaker models, as a large angle of feed will cause the paper to bunch up at the edge of the machine, causing a jam. Card stock, photographs, and light chipboard should be fed in single sheets, and the weakest models sometimes cannot handle staples.
Stronger models are available that can handle larger stacks of paper, up to 25 sheets at a time if you're willing to pay for that kind of power. Stronger models are also capable of shredding more than just paper. They can handle staples, paper clips, credit cards, and even CDs and DVDs. For truly massive shredding operations, however, commercial shredding companies would be happy to toss entire cardboard boxes full of documents into evil looking machines with huge spinning blades capable of shredding small trees... for a reasonable fee. This should be done on-site so you can visually verify the shredding is taking place.
An enormous variety of paper shredders are available, ranging from $20 to over $2000. Depending on how much you want to spend, you can get speeds up to 25 feet per minute, enough power to deal with paper clips, catch bins up to 30 gallons, and a throat of up to 16 inches wide to deal with 132-column continuous feed computer paper. The cheapest models run when turned on until turned off, but most come with an automatic mode that turns the motor on when paper is fed into the throat. The most popular kinds have a four position switch: On, Auto, Off, and Reverse for clearing jams.