Error coins are a numismatist's dream. In fact, some coin enthusiasts will get boxes or rolls of coins from banks or buy large unsorted coin collections just to carefully pick through them looking for errors. And that can be a worthwhile hobby - some error coins can sell for tens of thousands of dollars, and even if yours isn't super rare, it's still probably worth a lot more than face value.
The only true error coins are those that come out of the U.S. Mint with flaws. Some errors that can be found on coins are double struck (causing numbers, lettering, etc. to appear doubled), missing rim, off-center images, partial blank planchets (plain metal disks), clipped planchets, and more. Because of their value, fraudsters try to create an appearance of an error on a coin that was minted without errors. Let's take a look at some pennies that are known to have been faked - or counterfeited.
One of the most popular double struck (sometimes called "double die") pennies is a 1955 wheat penny minted in Philadelphia. About 40,000 of these pennies were created during one shift at the mint, when a misaligned die created the double strike on the front of the coin, most noticeable on the numbers and lettering. With values ranging into the thousands for a single one of these pennies, it's highly desirable and, therefore, highly counterfeited. Among the things to look for in a counterfeit is flat, squashed-looking lettering, thickness of some letters, spacing between "In God We Trust" and/or "Liberty" and the rim, weak lettering and less detail on Lincoln's image.
In 1943, during World War II, copper was needed for the war effort, so for a short time, pennies were made from steel instead of copper. However, a small number of 1943 copper pennies were minted, mostly in Philadelphia, and today only a handful are known to remain. These rarities are worth tens of thousands to hundreds of thousands of dollars. One exception was a one-of-a-kind 1943D penny mistakenly minted from a bronze planchet that sold in 2010 for $1.7 million. Of course, many fraudsters try to fake their way to a rare 1943 penny, sometimes by coating a steel penny in copper. You can easily determine if the coin is steel underneath by using a magnet - steel will react, whereas copper will not.
Some fraudsters will try to create an error coin by sandwiching two pennies in a vise and crushing them together until they make impressions in each other's surface. The resulting coins will have a second impression pressed into the original raised image. You may even be able to see the rim of the other coin pressed into the front or reverse. Impressed images are dead giveaways - when coins are minted, the images, lettering and numbers are raised, not impressed. If the coin had been struck a second time at the mint with a different or off-angle image, both images would be raised. A penny with an impressed second image is most likely a fake error coin.
A relatively recent (but made to appear circulated) fake error coin being made overseas is the 1951D wheat penny. This counterfeit coin shows an off-center image, but here's the key - all of these coins are absolutely identical to each other. When coins are erroneously struck off-center by the U.S. Mint, there aren't hundreds that come out exactly the same way. So if you find one of these 1951D pennies, compare it to images of known counterfeits before buying or ask knowledgeable coin collectors in online coin forums.
Coin collectors debate what the fate of these error coins should be - should they be destroyed so they won't fool anyone into spending their hard-earned dollars on a fake? Should they be put back into circulation? Many seem to agree that the best practice is to keep fake error coins in marked coin holders as examples and use them to help educate others who may be considering purchasing or selling a similar fake error coin.
When buying error coins, it's best to stick with coins that have been authenticated by grading services like PCGS or NGC. If it's ungraded, make sure you can return the coin to the dealer if it's proven a fake.

Photos via Wikimedia Commons.

At A Glance
Signs of a counterfeit or fake:
1. Off-center 1951D should raise caution.
2. Flat, squashed-looking lettering.
3. Impressed images.
4. Copper-clad steel pennies.

Reference books, websites and collecting groups are the best ways to keep from being fooled by fakes. Here are a few: “Lincoln Cent Error Coin Guide” by Stan McDonald; American Numismatic Association Coin Club Directory (https://www.money.org/club-directory); Coin Collecting Meetups (https://www.meetup.com/topics/coincollecting); and Coin Community (http://www.coincommunity.com/forum).