An infection is when something other than the desired yeast that you're pitching begins to ferment your batch. It could be another yeast, like brettanomyces, or it could be bacterial, like lactobacillus. They usually lead to off flavors which can be downright vomit inducing, but they won't kill you. Some styles rely on "infections" to make them what they are! Ever have a sour ale?
Another problem I have read about is carbonation: if you have an unknown bacteria producing unknown amounts of co2, it is easier to create a "bottle bomb", where the bottle gets over-carbonated due to the infectious bacteria.