das Mitglied
member
A person who belongs to a group, club, organization, or family.
Jedes Mitglied des Vereins muss einen Jahresbeitrag zahlen.
Every member of the club has to pay an annual fee.

member
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member
A person who belongs to a group, club, organization, or family.
Jedes Mitglied des Vereins muss einen Jahresbeitrag zahlen.
Every member of the club has to pay an annual fee.

The word consists of 'Mit-' and 'Glied'. 'Mit' means 'together/with', and a 'Glied' is a part of a whole (like a link in a chain). So a member is literally a 'part that belongs to it'.
Als neues Mitglied fühlte er sich schnell in der Gruppe willkommen.
As a new member, he quickly felt welcome in the group.
Although 'das Mitglied' (member) refers to people (male, female, diverse), the word is grammatically neuter. It always remains 'das Mitglied', regardless of the person's gender.
Sie ist seit zehn Jahren Mitglied in unserem Chor.
She has been a member of our choir for ten years.
You are 'Mitglied in' (a member in/of) a group in which you are active (e.g., a club), or 'Mitglied von' (a member of) a larger organization. Often both prepositions are possible.
Er ist Mitglied im Schachklub.
He is a member of the chess club.
Common verbs used with 'Mitglied' (member) are 'Mitglied werden' (to become a member), 'Mitglied sein' (to be a member), and 'als Mitglied austreten' (to resign/withdraw as a member).
Möchtest du Mitglied in unserem Fitnessstudio werden?
Would you like to become a member of our gym?
The meaning 'body part' is outdated (archaic). You will only find it in very old literature. It is no longer used in modern language.
Der Ritter schützte seine Mitglieder mit einer Rüstung.
The knight protected his limbs with armor.
man
'der Mann' and 'das Mitglied' can both refer to a person in a group or team. 'Mitglied' is used when someone belongs to a club, team, organization, or family. The word is neutral and works for all people. 'Mann' fits more when talking about people assigned to a team or task, for example 'zehn Mann' (ten people). This can mean males, or it can sound general in fixed expressions. The words can be confused because both occur in group contexts. When belonging is important, 'Mitglied' is more precise.

Jedes Mitglied des Teams bekommt vor dem Projekt eine kurze Aufgabe.
Each member of the team gets a short task before the project.
Für den Umzug brauchen wir noch zwei Mann.
We still need two more people for the move.