A ‘logical fallacy’ is an error in reasoning. That is, an argument that might have a pattern that is familiar, but that does not hold up to close scrutiny.
Here’s a crazy example:
My dog likes to chase the postman. Therefore my dog likes the postman.
If you know anything about dogs, you know that they aren’t chasing the postman because they ‘like’ him or her – they want to bite the postman.
The argument that the person’s dog likes the postman is a logical fallacy because it jumps to a ‘hasty conclusion.’ There are many, many reasons why a dog would chase a postman, or anyone else for that reason. Maybe the postman has a dead cat in his bag (you get the idea).
There are many different types of logical fallacies, and some people like to name them. For instance, “hasty generalization” is a category of logical fallacy. Teachers might give you the name of a fallacy you use, and you can find many terrific resources on the web to describe them in detail.
Here’s a good resource that helps you identify when people are using fallacies against you, and how to counter them.
This is a more complicated example, and at the end of the day the devil is in the details. That is, you could probably make an ethical, logical case for Sarah Palin’s daughter’s actions being a reflection on Sarah Palin’s family values and leadership abilities. But you’d have to do it carefully and with a lot of nuance.