As with the previous games in the series, Dark Souls 3 features a number of different endings depending on what actions you take during your playthrough. There are four of them in total and they're increasingly more complicated to unlock, so check out the requirements below to receive your outcome of choice.
Ending 1 - To Link the First Flame
This is the standard ending that most players will get, as it requires no additional quests to reach. Once you have defeated the Soul of Cinder, simply walk over to the bonfire and interact with it to trigger the final cutscene.
Ending 2 - The End of Fire
To receive this second ending, there are a couple of items you need to obtain. The first is the Fire Keeper Soul, which is found at the top of the tower behind Firelink Shrine. To reach it, you must purchase the Tower Key from the Shrine Handmaid for 20,000 souls.
The other item required is the Eyes of a Fire Keeper, which can be found in the Untended Graves hidden area. This location includes a 'dark' replica of the Firelink Shrine, in which you should head to the lower catacombs then clear the illusory wall at the end of a closed corridor to reveal the Eyes.
Give both of these items to the Fire Keeper at Firelink Shrine, then go through all of her dialogue options. With that done, defeat the Soul of Cinder then use the summon sign to bring in the Fire Keeper and watch the final scene play out.
Ending 3 - The End of Fire, Alternative
If you follow the steps for Ending 2 above, but attack and kill the Fire Keeper after summoning her, you'll initiate an alternative outcome. Note - you will NOT receive "The End of Fire" trophy/achievement for this ending.
Ending 4 - The Usurpation of Fire
To get this final ending, there are plenty of steps you need to take through your entire playthrough so make sure you're keeping track. Below the Foot of the High Wall bonfire, find Yoel of Londor among the fallen pilgrims and complete his dialogue to warp him to Firelink Shrine. Find him there and choose the option to "Draw Out True Strength" to level up, then repeat this every two deaths until you've levelled up five times, ensuring you complete this before killing the Abyss Watchers.
Next, locate Anri of Astora by the Halfway Fortress bonfire and complete their dialogue. After defeating the Deacons of the Deep but before killing the Abyss Watchers, return to Firelink Shrine and talk to them again. Meet them once more in the Catacombs of Carthus and complete their dialogue, selecting "no" and "do nothing" where appropriate - do not reveal Horace's location to Anri.
Speak to Anri when you reach the wooden bridge before fighting High Lord Wolnir, then chop down the bridge to enter Smouldering Lake. Here, stick to the right wall until you find a passage with a couple of Crystal Lizards in it, then head to the back of the area and kill Horace - but make sure you don't tell Anri what you've done. You'll see them next at the Church of Yorshka bonfire. Complete all their dialogue here, and do not kill the assassin who's hiding in the corner of the room.
After killing Pontiff Sulyvahn, return to Firelink Shrine and speak to Yuria until you've completed her dialogue. If you've done all the right things up to this point then she'll tell you to go to the Darkmoon Tomb where your spouse is waiting for you, so head down there (it's at the bottom of the spiral stairs leading to Anor Londo) and collect the sword from the pilgrim by the entrance to complete the 'wedding' ceremony.
When this is done, speak to Yuria once more at the Firelink Shrine, where she'll talk to you about the ceremony and your vows. Make sure you don't cure your dark sigil, then continue as before to defeat the Soul of Cinder and interact with the flame to receive the final ending.