Birth:
* The hospital celebrated the birth of its 1000th baby this year. (refers to the act of being born)
* She was born on a Tuesday, but her birth certificate says Wednesday. (refers to the official record of being born)
* The birth of the internet revolutionized communication. (refers to the beginning of something)
Berth:
* The ship docked and secured its berth at the pier. (refers to a designated space for a ship to moor)
* The train had to wait for a berth to become available before it could depart. (refers to a designated space for a train to stop)
* He booked a lower berth on the sleeper train. (refers to a bed on a train)
Remember that "birth" is always a noun, while "berth" can be either a noun or a verb.