(Answering because a lot of the other answers missed other very, very important information or used problematic language.)
Queer - a blanket term that covers anyone not heterosexual, heteromantic or cisgender. However, it was at one time, a slur for the LGBTQ+ community, so you should not use this term unless you are absolutely sure the person/group of people you are referring to are okay with it. You can call yourself queer, especially if you have a lot of labels (or you can't describe your gender/sexuality.)
Intersex - A person who does not conform to the general rules of "sex" (breasts + vagina, penis, etc). Intersex people can have combinations of different chromosomes plus the general unassociated-to-said-chromosome-genitalia, and some have internal organs that differ from the outside appearance of their genitals.
NOTE: Intersex people, at one time, were referred to as "hermaphrodites", which is an INCREDIBLY OFFENSIVE TERM, and should NEVER BE USED TO REFERENCE AN INTERSEX PERSON.
Transgender - A person who does not agree to their AGAB (Assigned Gender At Birth). For example: I am an AFAB (Assigned Female At Birth) Nonbinary (Meaning I do not conform to binary genders - male and female) person, meaning I identify as transgender. There is a common misconception that you can only be trans if you go from male to female or female to male. This is not the case, and it also is very uninclusive of intersex people (often times intersex people are assigned male or female at birth and don't know they're intersex until they're older). As mentioned above, a large portion of transgender people are AMAB (Assigned Male At Birth) or AFAB, but are ACTUALLY the opposite gender.
It is important to note that if you are AMAB, you don't have "a male body" with a "female brain", but rather your body is female, because you identify as female (and vice versa).
Often times, people use the word "transsexual" to refer to transgender individuals. Transsexual is a term that is perfectly okay to use, but has fallen out of use, and many transgender people find it offensive. Unless a person refers to themself as transsexual FIRST, do not call a transgender person transsexual. Also, transsexual can only be used if a part of your body has been altered to match your gender.