die Verbindung
connection
A technical connection for communication, data transmission, or electricity.
Meine Internetverbindung ist heute sehr langsam.
My internet connection is very slow today.

connection / association
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connection
A technical connection for communication, data transmission, or electricity.
Meine Internetverbindung ist heute sehr langsam.
My internet connection is very slow today.

'Die Verbindung' (connection) is the noun related to the verb 'verbinden' (to connect). The basic idea of a link or relationship appears in all meanings, whether technical, social, or logical.
Können Sie mich bitte mit der Personalabteilung verbinden?
Could you please connect me with the HR department?
In technical contexts, fixed combinations are very common. These include 'Internetverbindung' (internet connection), 'WLAN-Verbindung' (Wi-Fi connection), or 'Telefonverbindung' (phone connection). Typical verbs are 'eine Verbindung herstellen' (to establish a connection) or 'die Verbindung ist abgebrochen' (the connection has been lost).
Ich kann keine E-Mails senden, die Internetverbindung ist unterbrochen.
I can't send emails; the internet connection is down.
When talking about means of transport, the route is often described with the prepositions 'von' and 'nach' (from and to) or 'zwischen' and 'und' (between and and). For example, you might ask: 'Gibt es eine Verbindung von Köln nach Berlin?' (Is there a connection from Cologne to Berlin?)
Die Zugverbindung zwischen den beiden Städten ist sehr gut.
The train connection between the two cities is very good.
The phrase 'sich mit jemandem in Verbindung setzen' (to get in touch with someone) is a fixed expression that means 'jemanden kontaktieren' (to contact someone). Note that the person or office being contacted is always in the dative.
Ich werde mich nächste Woche mit dem Vermieter in Verbindung setzen.
I will get in touch with the landlord next week.
The expression 'in Verbindung mit' (in connection with) is a fixed prepositional phrase that means 'zusammen mit' (together with) or 'in Kombination mit' (in combination with). The noun that follows it is always in the dative.
In Verbindung mit Sport ist eine gesunde Ernährung besonders wichtig.
In connection with exercise, a healthy diet is especially important.
When referring to social or professional contacts, the plural 'Verbindungen' (connections) is often used. It usually refers to useful or influential contacts.
Er hat den Job nur durch seine guten Verbindungen bekommen.
He got the job only through his good connections.
This meaning is limited to the scientific field. If you see words like 'chemisch' (chemical), 'Element' (element), 'Stoff' (substance), or a chemical formula, a chemical compound is meant.
Kochsalz ist eine chemische Verbindung aus Natrium und Chlor.
Table salt is a chemical compound made of sodium and chlorine.
In Germany and Austria, 'die Verbindung' can also refer to a traditional student association at a university. This term is very specific to the university context.
Er ist während seines Studiums einer alten Verbindung beigetreten.
He joined an old student association during his studies.
ribbon / band
'das Band' and 'die Verbindung' can both mean a relationship between people. 'das Band' sounds more emotional and close, for example between family members or good friends. 'die Verbindung' is more general and can also mean a social or professional contact that is useful. Learners may confuse the words because both describe something that connects people with each other. When feeling and closeness are important, 'Band' fits better. When it is about contact, a relationship, or usefulness, 'Verbindung' is more natural.

Durch seinen alten Chef hat Paul eine gute Verbindung zu der Firma.
Through his old boss, Paul has a good connection to the company.
Zwischen den Geschwistern besteht ein starkes Band.
There is a strong bond between the siblings.