To connect or link things together, or to join a relationship/flow (can be physical or abstract).
ケーブルをテレビとレコーダーに繋いでください。
Please connect the cable to the TV and the recorder.
この橋は二つの島を繋いでいる。
This bridge connects the two islands.
新しいSNSは世界中のユーザーを繋ぐことを目指している。
The new social network aims to connect users around the world.
異文化交流が人々の心を繋いだ。
Cross-cultural exchange brought people's hearts together.
To tie, fasten, or secure with a string/rope; can also mean to restrain.
犬を外の柵に繋いでおいてください。
Please tie the dog to the outside fence.
荷物が動かないようにロープでしっかり繋いだ。
I tied the luggage tightly with a rope so it wouldn't move.
幼い子の手をしっかり繋いで歩いた。
I held the small child's hand firmly as we walked.
To maintain or preserve something so it continues without interruption (e.g., keep life, preserve traditions).
その薬は彼の命を繋いだ。
That medicine kept him alive.
伝統を次の世代に繋ぐことが大切だ。
It's important to pass traditions on to the next generation.
資金を繋ぐために短期の融資を受けた。
We took a short-term loan to keep funds available.
To transfer a phone call or put someone through to another person/department (polite: お繋ぎします).
少々お待ちください、担当者にお繋ぎします。
Please wait a moment; I'll put you through to the person in charge.
この番号にかけ直せば、受付につながります。
If you call this number back, you'll be connected to reception.
To hedge or lock in a price (e.g., by forward/derivative contracts) — a specialized financial usage.
為替リスクを避けるために、円のレートを先物で繋いだ。
To avoid exchange-rate risk, we hedged the yen rate with a forward contract.
In the game of go: to connect stones so they support each other (also written ツぐ or ツナギ in related terms).
この一手で黒の石を繋いだ。
With this move, Black connected his stones.