It's easy to think of our minds as being identical with our consciousness. Easy because, by definition, we're not aware of thing things we're not conscious of.
However, we know that we have lots of unconscious processes going on. For example, when you talk to someone, how to you come up with the exact words your'e going to use? When you see your friend, and recognize her face, how do you know it's her? Reflecting on your own mind does not provide any guide to how these things work. It's even misleading to say it feels easy--it doesn't feel like anything at all! But it's a complicated cognitive feat that our consciousness is just not privy to.
For cognitive scientists to get computers to do good face recognition took about fifty years of research, and we still are having a hard time getting computers to generate decent sounding sentences (and I'm only talking about the text, let alone realistic sounding speech).
Sometimes these unconscious processes come up with ideas about perceptions. Take the image presented. This is a famous optical illusion which appears to be moving. It's not an animated GIF or anything-- if you stare hard at any one part, you will see that it's not moving, while the rest of it (the parts you're not focusing on) seems to move in your peripheral vision. This kind of effect is typical of optical illusions.
Okay, so let's assume that you have convinced yourself that the image is not actually moving. Does the illusion go away? No, it still looks like it's moving! How is it that you can see it moving but know that it's not?
This problem is not restricted to optical illusions. You might find yourself not trusting someone you meet, and then realize it's just because they remind you of someone who wronged you in the past. Even though you know better, you can't shake the feeling of mistrust. Or perhaps you have the opposite reaction to someone who reminds you of a past lover, and you feel an unjustified intimacy with that person. When you watch someone on film, you can't help but see them as a person, even though you might know that it's just light on a screen. It's hard to see it as light on a screen though, and impossible to not perceive a face.
How is this possible? The answer is that our minds are complicated, and have a great deal of cognitive processes that can sometimes work independently of the other ones. Optical illusions work by taking advantage of particular aspects of your visual system that your conscious mind cannot just turn off. In the case of the optical illusion presented, your motion detecting processes are at work, telling you (you being your conscious mind) that there's motion there, and your more deliberative processes, well, they just disagree. You are literally of two minds.
Part of your mind can believe one thing, and another part another.
If you identify your self primarily with your conscious mind, you could say that you don't believe one of your mind's beliefs.
What To Call Them
In cognitive science there is no good name for what I've been calling "beliefs," and the above argument shows you why the term "belief" isn't great. Saying that sometimes beliefs are not believed sounds kind of silly, because the common idea of a belief is that it's something you endorse. But of course we have...things in our minds that we don't endorse.
Take the idea that the Earth is flat. We don't believe it, but we've all heard of it. We remember it. We have it represented in our minds, but we also have represented, somehow, that we don't believe it. We believe that the Earth is (roughly) spherical, and we have stored in our minds the non-endorsed idea that the Earth is flat that we heard in a history class. Just because we don't believe it doesn't mean we forgot it.
So the term belief doesn't really work because we don't necessarily believe them.
The term fact doesn't work because facts need to be true, and we can believe and represent untrue things.
The term memory is too broad, as memories include procedural things such as how to snap your fingers, which are neither believed nor not believed.
The term proposition is a little better, but in philosophy that term is defined as having a "truth value" of either true or false. In cognitive science we'd think of it as "believed" or "not believed," but even that is an oversimplification, because evidence shows that we evaluate truth on a continuous (or at least not a binary) scale (Hampton, 2007). That is, a man who is 5'9" is not clearly tall and not clearly not tall. He's kind of tall. So is "The man is tall" true or false? It's somewhere in between. This is actually really common.
The term sentence is no good because it implies that it's represented in some natural language, such as English. For example, you believe that your mother looks like she does, but this is not something that can be expressed as a sentence.
The term knowledge also has a philosophical meaning which would provide unwanted baggage. Philosophers consider knowledge to be a subset of beliefs that are true and that the agent is justified in believing. So it suffers from the same problem as fact.
Perhaps the best word is idea, though it has strong connotations with novelty and creativity.
Who cares what we call them? Perhaps it's just a semantic issue. But calling them any one of the terms above limits how we tend to think about them, and forget, for example, that we don't even believe all of our beliefs.
Hampton, J.A. 2007. Typicality, Graded Membership, and
Vagueness. Cognitive Science, 31 (3), pp 355-384.
For more optical illusions, see http://www.michaelbach.de/ot/