Bloodborne has three distinct endings, one of which requires a little extra work to trigger at the end of the game. In this guide I'll describe how to watch each of the endings. Please note that you can only view one ending per playthrough unless you use an external USB device to create a save file on.
The 'True' Ending: Childhood's Beginning
After killing Mergo's Wet Nurse you'll need to acquire three pieces of the One Third of Umbilical Cord, identified as "A great relic". There are actually four pieces available in the game, so you can skip one. Their locations are the following:
1. Mergo's Wet Nurse drops the first One Third of Umbilical Cord. This piece is mandatory as she is not a skippable boss.
2. One of the pieces requires that you spoke to Arianna near the Cathedral Ward lamp earlier in the game. If you sent her to Oedon Chapel, you will find a blood trail at her location next to the lamp upon defeating Mergo's Wet Nurse. Follow the blood trail and you will find a dead Arianna and a demon baby. Killing the demon baby will award you with a One Third of Umbilical Cord.
3. Visit Iosefka's Clinic where the game started by teleporting to Central Yharnam, turning around, and going down the ladder before heading into the graveyard to your left. The clinic will be to your right upon entering this area. Go up to the door at the end of this building to confront Iosefka. Upon defeat she will drop a One Third of Umbilical Cord.
4. You will need to head to the Abandoned Old Workshop. To get there spawn at the Cathedral Ward and immediately go right. Take the elevator up, go left, and then drop down the large hole in the floor to your right. You'll then need to navigate the wooden scaffolding and drop down to a landing point about halfway down where a door to the Abandoned Old Workshop is. If you're having difficulty seeing use a torch. The One Third of Umbilical Cord piece is a lootable item in the Abandoned Old Workshop.
Once you have three of the four One Third of Umbilical Cords, consume them using the action in your inventory. Then, head to Hunter's Dream and speak to Gehrman. Refuse his offer and you can view this ending.
The 'Good' Ending: Honoring Wishes
When speaking to Gehrman, the First Hunter after killing Mergo's Wet Nurse, choose to accept his offer. You will then need to battle him before this ending will play. Gehrman is a very difficult encounter that is best done if you are level 90 or higher.
The 'Bad' Ending: Yharnam Sunrise
When speaking to Gehrman, the First Hunter after killing Mergo's Wet Nurse, choose to reject his offer. This will skip the battle with him and immediately trigger this ending.