What are the parts of a name?

For example, in Latin America the full name would be like:

Ana María Gómez Sánchez

In that case, "Gómez" is her dad's last name, and "Sánchez" is her mum's single last name.

Let's pretend Ana María's dad's name is Gabriel Gómez Pérez. And her mum's name is Jessica Sánchez Flores. That way, Ana María gets both of those first last names.

What I want to know is what's the name of every part of the name? What's the last name, the middle name, the surname, and if there's more, haha.

Sorry if it's confusing, and thank you in advance!

Since you asked, here are the most of the possible parts of a name in English from my (BrE) perspective: You can skip to the bit about other languages if you wish; P.S> you seem to be asking about English and other languages(?)

First name (forename):

Pretty self-explanatory; the first part of someone's name. When you are informally addressing someone known, you will typically only use their first name.

John Smith

Last (family) name:

Also straightforward, the last part of someone's name, in most cases taken from their father (their family name). You will typically introduce someone with their first and last name in a formal situation. The first and last names of a person are generally called their full name, though this sometimes includes their middle name(s)^{Lexico Oxford Dictionaries}. In your case, the "Sánchez" part would be in English referred to as a maternal or matronymic surname. There's a lot more on this here.

John Smith

Middle name(s):

These are the names that go in-between the first and last names, and are usually taken from grandparents/ancestors etc. A person can have several middle names.

John Edward Lawrence Smith

Nickname or moniker: These informal titles are almost exclusively used in conversation between friends, and pretty much never formally. They are styled in writing usually like this:

John "Johnny" Smith

Honorary and Professional Titles:

You achieve these titles either through gaining a degree. First of all, a professional title is achieved through education (like a P.h.D [Doctorate] or MA[Master of Arts]). A knighthood or, in the UK, something like an Order of the British Empire (OBE) is achieved through services to your country. A hereditary title is inherited usually through parents e.g. "Count" or "Duke".
Note that if you are a doctor and are knighted, you would put the Sir at the beginning and the "Dr." would become "P.h.D" at the end:

Sir John Smith, P.h.D

In the same way, Count or some such takes precedence over Sir. There is more on this that is not of significant relevance here at Debrett's (via WayBackMachine)

There is loads more on complicated English naming customs than is appropriate here at Wikipedia.

In other languages,

Spanish, as in your case:

a person's name consists of a given name (simple or composite) followed by two family names (surnames). Historically, the first surname was the father's first surname, and the second the mother's first surname.

If you're very interested in the complications of the Spanish naming system, there's a very long and detailed description at Wikipedia, which listing here would be inappropriate: Spanish naming customs

To keep it simple and comprehendible:

All names taken from the mother are matronymic or maternal All names taken from the father are patronymic or paternal

So, in an English description

Ana María is her personal forename (See my part about English above; this is how it links)
Gómez is her paternal surname(taking precedence over the maternal one
Sánchez is her maternal surname.

Hope this helps.