badge / marking
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badge
A small symbol, often worn on clothing and made of fabric, metal, or plastic, that indicates membership, an achievement, or a particular function.
Nach der bestandenen Prüfung erhielt sie stolz das Schwimmabzeichen.
After passing the exam, she proudly received the swimming badge.
The word is made up of the prefix 'ab' and the noun 'Zeichen' (sign/mark). An 'Abzeichen' (badge/marking) is therefore literally a sign that sets someone or something apart from others or marks it in a special way.
Für seine Tapferkeit erhielt der Soldat ein besonderes Abzeichen.
For his bravery, the soldier received a special badge.
In connection with achievements or membership, an 'Abzeichen' (badge) is often 'verliehen' (awarded), 'getragen' (worn), 'erworben' (earned), or 'angesteckt' (pinned) to clothing.
Sie trägt das Abzeichen ihres Vereins an der Jacke.
She wears her club's badge on her jacket.
You can recognize this meaning from the context: when it is about animals, especially horses or dogs, 'Abzeichen' (marking) describes a natural colored marking in the fur.
Das schwarze Pferd hat ein kleines weißes Abzeichen am Bein.
The black horse has a small white marking on its leg.
sign / symbol
'das Zeichen' and 'das Abzeichen' can both be a visible symbol. 'Zeichen' is the general term for a sign, symbol, or marking. 'Abzeichen' is more specific: it is usually a small symbol worn on clothing that shows membership, a task, or an achievement. The words can be confused because an 'Abzeichen' is also a kind of 'Zeichen'. When talking about a symbol in general, you say 'Zeichen'. When someone wears something on their clothing, for example for a club or after an exam, 'Abzeichen' is more precise.
Nach dem Schwimmkurs bekam Lisa ein Abzeichen für ihre gute Leistung.
After the swimming course, Lisa received a badge for her good performance.
Dieses Zeichen auf dem Plan zeigt, wo der Treffpunkt ist.
This symbol on the map shows where the meeting point is.