Most of the time, the German word for “connected” is verbunden. It can be used to describe a connection to the internet as well as to express an association with other people, or being connected on the phone.
The word angeschlossen is very similar, but used somewhat more in the sense of “attached”, “plugged in” or “joined”.
Examples
Bist du mit dem Internet verbunden? Are you connected to the internet?
Er war mit dem Verein verbunden. He was connected to (or affiliated with) the association.
Ich habe mich der Partei angeschlossen. I joined the (political) party.
Du musst das Kabel anschliessen. You have to connect (or plug in) the cable.
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Nouns
The German nouns for “connection” are Verbindung (f.) or Anschluss (f.):
Wir haben keine gute Verbindung mehr. We no longer have a good connection.
Ich habe den Anschluss verpasst. I missed the connection.
Another Meaning
For “connection” in terms of “context”, the word is Zusammenhang (m.):
Ich sehe den Zusammenhang nicht. I don’t see the connection.
For “connection” in terms of a “relationship”, the word is Beziehung (f.). A “connecting flight” is Anschlussflug (m.).