Depression is a mood disorder. That is, it puts your feelings into disorder by taking what a person normally feels briefly when they are particularly sad over something, and locks that mood in place and stretches it out over weeks or even longer.
Anxiety is similar in that it causes the sufferer discomfort by taking what a person naturally feels when they are physically in real and present danger and locks that state of mind and body in place when the person is not in any danger. This can be over an extended period of time in a like manner to how depression works, or come upon the sufferer briefly and suddenly in the case of conditions such as panic disorder.
Psychosis is a state of mind where the sufferer cannot separate the real from the imagined. This can take the form of hallucinations or delusions. Hallucinations are real-to-the-sufferer sensory experiences involving seeing, hearing, smelling, tasting or touching something that is not actually there. Delusions are real-to-the-sufferer beliefs about themselves, others and the world around them that run contrary to good sense and reason. Psychosis is common in people with schizophrenia and bipolar disorder type 1.
It is important to remember that mental illness is a very personal thing: that is, it differs from person to person and expresses itself uniquely in each person.