My biggest pet peeve is bad grammar, written or spoken. Even the smallest mistakes — such as the objective use of the word "who" or no comma after a subordinate clause that precedes an independent clause — get on my nerves far too much.
Another huge pet peeve of mine is when people say or imply that humans are not animals. The notion that humans are not animals is extremely ignorant and outdated, and there aren't any decent arguments to support it.