A bridge is what gets you from one side of the river to the other. A bridge connects things. The bridge of your nose connects your nose to your head — it's where the bridge of your eyeglasses rests.
If you insult your boss and quit your job, you've "burned your bridges" — you can’t go back. If you're not going to worry yourself about a possible problem, you can say "I'll cross that bridge when I get to it." The vertical piece that holds the strings up on a violin, guitar, or other stringed instrument is also a bridge, and in a song, the bridge is the part between a verse and the chorus — the transition between them.