die Karte
card
A small, usually rectangular piece of cardboard or plastic for a specific purpose (e.g., postcard, playing card, credit card).
Kannst du bitte mit der Karte bezahlen?
Can you please pay by card?

card / map
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card
A small, usually rectangular piece of cardboard or plastic for a specific purpose (e.g., postcard, playing card, credit card).
Kannst du bitte mit der Karte bezahlen?
Can you please pay by card?

Because 'Karte' (card/map/menu/ticket) has so many meanings, the exact meaning is often clarified by a compound word. Pay attention to words like 'Kreditkarte' (credit card), 'Landkarte' (map), 'Speisekarte' (menu), or 'Eintrittskarte' (admission ticket).
Hast du die Eintrittskarten für das Museum schon online gekauft?
Have you already bought the admission tickets for the museum online?
For cards made of plastic or paper, typical verbs are 'bezahlen' (to pay) (with a credit card), 'schreiben' (to write) (a postcard), or 'ziehen' (to draw) and 'spielen' (to play) (with playing cards).
Beim Kartenspielen hat er die beste Karte gezogen.
While playing cards, he drew the best card.
When 'Karte' means a map, the preposition 'auf' (on) is often used. Typical verbs are 'lesen' (to read), 'studieren' (to study), or 'nachsehen' (to check/look up).
Wir müssen auf der Karte nachsehen, wie wir zum Hotel kommen.
We have to check on the map how to get to the hotel.
In a restaurant context, 'die Karte' almost always means the food or drinks menu. A typical question to the waiter is: 'Könnten wir bitte die Karte haben?' (Could we please have the menu?)
Ich habe etwas Leckeres auf der Karte entdeckt.
I discovered something delicious on the menu.
The idiom 'gute Karten haben' (to have good cards/chances) means that someone has good chances or prospects of success. The person who has the good chances is the subject of the sentence.
Mit ihrer Erfahrung hat sie für den Job sehr gute Karten.
With her experience, she has very good chances for the job.
The idiom 'alles auf eine Karte setzen' (to put everything on one card) means that you take a high risk for a single possibility. It is often used in financial or professional contexts.
Er hat seinen Job gekündigt und mit seinen Ersparnissen alles auf eine Karte gesetzt.
He quit his job and put all his eggs in one basket with his savings.
plan
'der Plan' and 'die Karte' can both be drawings that help with orientation. For places in a city, people often say 'Stadtplan' (city map). 'Karte' is more general for a geographic representation, for example a 'Landkarte' (map) or 'Wanderkarte' (hiking map). 'Plan' can more strongly show the exact structure of something, such as buildings or routes in a city. The words are easy to confuse because in everyday life a city map is used like a map. Also important: 'Plan' can additionally mean an intention, but 'Karte' cannot.

Im Urlaub benutzt sie eine Karte, um mit den Kindern den See zu finden.
On vacation, she uses a map to find the lake with the children.
Nach dem Unterricht sucht die Klasse auf dem Plan den schnellsten Weg zum Museum.
After class, the class looks for the fastest route to the museum on the map.