das Glied
link
A single part of a larger whole that consists of several similar parts, such as in a chain or a series.
Ein schwaches Glied kann die ganze Kette reißen lassen.
One weak link can cause the whole chain to break.

member / limb / link
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link
A single part of a larger whole that consists of several similar parts, such as in a chain or a series.
Ein schwaches Glied kann die ganze Kette reißen lassen.
One weak link can cause the whole chain to break.

The different meanings of 'Glied' can be derived from the basic idea of 'part of a whole.' Related words are 'gliedern' (to organize something into parts) and 'das Mitglied' (a person as part of a group).
Die Firma ist in verschiedene Abteilungen gegliedert.
The company is divided into different departments.
The meaning 'male sexual organ' is very common in everyday language. Learners should be aware of this meaning to avoid misunderstandings when the word is used without clear context.
Der Arzt fragte nach Schmerzen am Glied.
The doctor asked about pain in the penis.
When people talk about 'den Gliedern' (the limbs, plural), they almost always mean the arms and legs. This form is often used in connection with pain, tiredness, or cold.
Nach der Gartenarbeit taten mir alle Glieder weh.
After gardening, all my limbs hurt.
The expression 'das schwächste Glied in der Kette' (the weakest link in the chain) is very common. It describes the most error-prone or weakest part of a system, group, or process.
Bei diesem Projekt ist die veraltete Software das schwächste Glied in der Kette.
In this project, the outdated software is the weakest link in the chain.
In the context of linguistics, 'Glied' or 'Satzglied' refers to a functional part of a sentence, such as the subject, predicate, or object.
In der Analyse bestimmen wir die einzelnen Glieder des Satzes.
In the analysis, we identify the individual elements of the sentence.
The meaning 'member of a community' is outdated today. It is found mainly in historical, literary, or very formal texts.
Er betrachtete sich als nützliches Glied der Gemeinschaft.
He saw himself as a useful member of the community.
part
'der Teil' and 'das Glied' can both mean an individual component of a larger whole. 'Teil' is much more general and fits almost any time something belongs to a whole. 'Glied' is used more when several similar parts are connected, for example in a chain, or when talking about arms and legs as body parts. For that reason, 'Glied' cannot simply replace every 'Teil'. Learners may confuse the words because both refer to something individual within a whole. If you are unsure, 'Teil' is usually more natural.

An der alten Kette ist ein Glied gebrochen.
A link in the old chain is broken.
Sie hat im Zug einen wichtigen Teil des Berichts gelesen.
She read an important part of the report on the train.