Hope this helps and thanks for the questions!
Non-binary refers to gender identities other than the traditional female and male "binary" identities.
In maybe more simple terms, a non-binary person is someone who does not identify as exclusively a man or a woman. Someone who is non-binary might feel like a mix of genders, or like they have no gender at all.
Queer is often used as a term for someone who is gay, but in recent years has become popular as an umbrella term for someone who identifies as anything other than straight/cisgender. Some people also choose the label ‘queer’ as a way of expressing themselves by their own personal feelings.
So, again in maybe simpler terms, and this is how I define them both, is queer essentially means ‘a person who is not straight’, while gay means ‘a person who is attracted to the same sex’. However it can be confusing because personally I am gay but since 'queer' also means not straight then I fall under that definition as well.
The 'Q' in LGBTQ can mean 'queer' or 'questioning' or both actually so you aren't wrong!