der Vorname
first name
A person's personal name that comes before the family name.
Bitte tragen Sie in das Formular Ihren Vornamen und Nachnamen ein.
Please enter your first name and last name on the form.

first name
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first name
A person's personal name that comes before the family name.
Bitte tragen Sie in das Formular Ihren Vornamen und Nachnamen ein.
Please enter your first name and last name on the form.

The word is built very logically: it consists of 'vor' (before) and 'Name' (name). So it is the name that comes 'before' the family name in the order.
Mein Vorname ist Alexander, aber die meisten nennen mich Alex.
My first name is Alexander, but most people call me Alex.
The word 'Vorname' (first name) is especially common in official contexts such as on forms, in ID documents, or during registrations. There, it usually appears right next to the field for the 'Nachnamen' (last name) or 'Familiennamen' (family name).
Füllen Sie bitte die Felder für Vorname und Geburtsdatum aus.
Please fill in the fields for first name and date of birth.
The phrase 'jemanden beim Vornamen nennen' (to call someone by their first name) means that you use informal address with a person and have an informal, familiar relationship with them. It is a sign of closeness.
In unserer Firma nennen sich alle Kollegen beim Vornamen.
In our company, all colleagues call each other by their first names.
name
'Name' and 'Vorname' both belong to naming a person. 'Name' is general and can mean the full name or, depending on the situation, only part of it. 'Vorname' is more specific: it means the personal name that comes before the family name, for example 'Anna' in 'Anna Müller'. Learners may confuse the words because in everyday life people often simply ask for the 'Namen' (name). But if a form asks for 'Vorname', you should not enter the last name or the full name.

Beim Termin trägst du deinen Vornamen in das erste Feld ein.
At the appointment, you enter your first name in the first field.
Die Mitarbeiterin fragt nach dem Gespräch noch einmal nach meinem Namen.
After the conversation, the employee asks for my name again.