Most psychological assessments are informal. The intake interview, for example, is a kind of assessment. Formal testing—including IQ tests, personality tests, and tests to identify disorders such as ADHD—is less common in therapists' offices than most people realize, mostly because insurance (at least in the US) often doesn't cover them, making them prohibitively expensive.
An objective personality test (in this case that means you answer true or false) with 567 questions. It's usually scored by a Scantron machine. The results can be submitted to a scoring service, which returns information on the scaled scores, along with an interpretation. Therapists always refer to this test as "the MMPI."
The MMPI is one of psychology's most used tests because of its rigorous validity and reliability. It also doesn't hurt that it has a Lie scale, a fake good scale, and a fake bad scale.
In spite of all of this, there are complaints about the test. For example, though Scale 5 (Masculinity/Femininity) does tend to generally to identify those who hold more masculine views (usually men and people with higher education, SES, and intelligence levels), it was originally intended to identify who was gay. It couldn't. Now that the psychological community has acknowledged that sexual orientation is normal regardless of which kind you have, the scale is usually just ignored.
The TAT is a subjective personality test. It consists of 31 black and white cards that have story images that can be interpreted in different ways by different people. The therapist chooses cards that seem most relevant to the client's issues rather than using all 31.
It's rarely used in modern American clinical settings because the scoring is almost as subjective as the test itself (ie it's not particularly valid or reliable, and many also argue that it's outdated). It also assumes white ethnocentrism; fortunately, tests like the TEMAS (Tell-Me-A-Story) are changing that. Intended for children, the TEMAS (said tee'-mass) consists of 23 color cards depicting ambiguous daily life situations in Black and Hispanic families and neighborhoods.
The TAT is referenced in Michael Crichton's Terminal Man and Thomas Harris' Red Dragon.
The Rorschach is the infamous "inkblot test." The Rorschach (pronounced roar-'shock) consists of 10 cards, 5 in black in white; 2 in black, white, and red; and 3 in full color. The inkblots are always the same, and all 10 are used. Rorschach interpretation is valid and reliable enough to be used in court if 1) the Exner scoring system is uses and 2) the therapist has had extensive training in the Exner system.