A mental state is a state which only a creature with a mind can be in. Propositional attitudes like beliefs and desires, pains, itches, and other bodily sensations, and perceptual experiences are examples of mental states.

A propositional attitude is an attitude one bears to a proposition (which is the sort of thing that can be true or false, and therefore can be believed or denied).  For example believing the proposition to be true, or disbelieving the proposition, or hoping that it won't come true, or desiring that it be true, is a propositional attitude.  So, "I fear that Joe might be sick" expresses a propositional attitude (fear) towards a proposition ("Joe might be sick")
Propositional attitudes are  intentional or representational states.
NOTE: The word "intentional" here is a term of art, and doesn't  especially refer to intentions or decisions, although, like beliefs and desires, intentions are also a kind of intentional state.

Intentional states have certain distinctive features:

1. Intentional states can be about things being one way rather than another. For instance, the belief that SIUE is in Edwardsville is about SIUE and its being in Edwardsville rather than elsewhere; my hope that I'll win the lottery is about me and my winning the lottery rather than not wining; my inteniton to go to Chicago is about me and going to Chicago rather than staying home or going to New York, etc..
2. Intentional states can represent things about an object, without there being any particular object they are about. For instance, I may desire to have a glass of water, without there being any particular glass of water such that my desire is for that glass of water.
3. Intentional states can represent things about an object that doesn't exist or even may not possibly exist.  Ponce de Leon believed that the Fountain of Youth was in Florida, although there isn't such thing.
4. The mere fact that an intentional state represents some object, and fails to represent it as being F, does not entail that the state represents the object as being not-F. There is a gap between "not-(representing it as F)" and "representing it as (not-F)." For instance, my memories of Fido do not represent it as having blue eyes. But they do not represent it as having non-blue (say, brown) eyes either. I just cannot remember what color its eyes were.
5. Intentional states can represent that a thing X is thus and so, and fail to represent that a thing Y is that way, even if X is Y. For example, Don Montero believes that Zorro is a daring and courageous man; he laso believes that Diego de la Vega is a timid man of little courage.  And yet, Zorro and Diego de la Vega are the same man.

There have been various attempts at finding a single feature or set of features that all and only mental states and processes have.  Here are some proposals and their problems:

• Mental states are intentional or representational.

Problem: Propositional attitudes and sensations are indeed intentional.  But there are mental states (e.g., pain) not reducible to these.  But what do our pains represent? And isn't it possible to be anxious and depressed without being anxious and depressed about anything?
• We have priviledged access to our mental states.  Your knowledge of your own mental states is not usually based on any evidence or observation. You can just tell that you're thinking about New York.  While other people do have to infer what you're thinking, from your behavior and from what you say, you don't have to. being one way rather than another.  You have a kind of special or privileged access to your mental states which other people lack.  By contrast, you don't have any such privileged access to the rest of the world, including your own body.  So we might take privileged access to be a mark of the mental.

Problem: It's is not clear whether we really do have privileged access to all of our mental states. For instance, other people might be in a better position than you are to know whether you're jealous of your friend's success, or what your unconscious motivations are.
• Mental states are conscious. There is some peculiar way it feels to be in those mental states, e.g., pleasure. However, there is no special way it feels to be 6 feet tall, on the other hand: the statue is 6 feet tall, but it doesn't feel anything. So we might take consciousness to be a mark of the mental.

Problem: We also seem to have many mental states which aren't conscious.