Certainty
Certainty is a topic that confuses a lot of people because they use an unrealistic standard. The typical problem is that they mean certain in a way that doesn't allow for any possibility of being wrong. The typical line is that if you can't know everything, you can't know anything. So as long as you don't know everything, you can't really be certain, can you? Obviously that kind of standard would be impossible to meet unless you were omniscient. And we're not.
So what does it mean in human terms? Well, first we have different degrees of confidence in our knowledge, right? We have varying amounts of evidence to support an idea. The evidence comes in different forms, and we have to identify how important each piece of evidence is. We do all of this in the full context of our knowledge up until that point. We decide how strong the evidence is, and whether there are reasons to doubt the validity of the conclusion. The stronger the claim, the more certain we are.
If we do think of it as a scale, what's on the high end? There's possible. There's probable. The term "certainty" is usually designated to be the highest level of confidence you can have. It amounts to saying that you have no (rational) doubts about it. Certainty is when all of the evidence available supports a particular conclusion, and no other conclusion can be found. Obviously if the data supports more than one theory, you wouldn't have grounds to be certain one is true. Similarly if the data contradicts the conclusion, you'd have a serious reason to doubt it.
Sometimes Objectivists call this "contextual certainty". The point is that it isn't some kind of certainty that is absolute, that ignores all future evidence, and has no possibility of being correct. Instead, it holds that you can be certain within the context of your knowledge. You treat the conclusion as if it were true, and you act according to it. If the context changes, your certainty can change as well. But until you find some reason to doubt the conclusion, you accept it as true and move on.
The contextual part of this is very important. You're not evaluating the validity of a statement in isolation from the rest of your knowledge. You're putting it in with everything else you know, and deciding that it is the only conclusion that fits the bill.
When people reject the possibility of certainty, they are forced to reject the possibility of knowledge as well. For them, you can never be sure what the relationship is between your knowledge and reality, so there's no reason to accept any of it as valid. The logical conclusion is an epistemological theory called Skepticism. Skepticism rejects the possibility of knowledge. It says simply that because we don't know everything, we can't know anything at all. In practice, this means every statement is equally worthless. Of course, nobody can practice this theory consistently, so instead it's used as a tool of evasion.
Some people have tried to take a soft-skepticism point of view by talking about "probabilistic certainty". It amounts to giving probabilities to your knowledge, and then when you form a conclusion based on it, you take into consideration the probabilities. So if you believe something at an 80% level, and then you base some conclusion on it, the conclusion would be no more than 80%. Something along those lines. Bayes Reasoning is a new and popular way of doing the math. But the problem is, what do those numbers mean? Where do you get them? Either you have knowledge of the world, or you don't. You can't have both. If your knowledge is not connected to reality, then how can you judge by how much it's off by? That would require knowing about reality. Probabilistic certainty is just another way of arguing against certainty.
Why would people argue against certainty? Some do it because they want an "out". If confronted with a contradiction or an ethical failing, they want to be able to point to the probability and say "We can't be sure!". Some do it because they think people with strong convictions are dangerous, and only doubt and uncertainty can keep people peaceful. Rand argued against this last one by saying that when a thug comes up and declares his own certainty, you can't beat him by expressing your lack of certainty.
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