What are the different endings of Sekiro?

There are 4 possible endings in Sekiro.

The biggest decision you make occurs right after collecting the Lotus of the Palace and Shelter Stone, and meet Owl on top of Ashina Castle. When you talk to him, you're given a choice to Follow the Iron Code or Serve Lord Kuro. Choosing to follow the iron code locks you into the Shura ending, while all 3 other endings remain available if you choose to side with Kuro.

Shura

When forced to choose sides, follow the iron code and side with Owl, and you'll end up with this ending. Note that this ending cuts off a significant portion of the game, but also allows you to challenge two bosses that are only available with this ending.

Immortal Severance

Choose to serve Lord Kuro, play through the rest of the game as normal, and give Kuro only the Divine Dragon Tears to trigger this ending.

Purification

This ending requires a bit of spying as you're required to eavesdrop on several characters to get this ending. After a bit of spying, you'll unlock Father's Bell Charm, which allows you to revisit Hirata Estate to fight Owl a second time. Doing this will reward you with the Aromatic Branch, which when given to Kuro at the end of the game (along with the Dragon Tears) will trigger this ending.

The required steps must be done before fighting the Divine Dragon or you'll be locked out of this ending.

Return

The most complicated ending to unlock, the Return ending requires you to finish a lengthy side quest for the Divine Child, which involves collecting several key items. Finishing this side quest rewards you with the Frozen Tears. Give these to Kuro at the end of the game along with the Dragon Tears, and the Return ending will play.

The steps for this ending must also be completed before fighting the Divine Dragon, else you'll be locked out.

Also related, Which bosses can only be fought in certain endings?

I'm going to open with the link to Fextralife, this site has the full list of all the requirements and steps needed to access all of the endings.

For the 'quick version', read below. Also... Spoilers? Although I would assume that is obvious since you are asking about endings.

At around the 2/3 (ish) part of the game, Sekiro will have collected a few Macguffin items and will need to return to his lord in Ashina castle. There are 2 specific items you need to acquire and will happen across all playthroughs. Once the second has been acquired, the game state will change and you can longer warp directly back to the castle, its under siege. You must climb your way back to the top, although its much easier this time around since 1) you've done it before and 2) the attackers have helpfully left walkways all over the place.

Shura vs. Non-Shura ending

At the top, your father will be waiting with a cutscene at which point you are presented with a choice. Follow the Iron Code or Betray the Iron code (Follow Lord Kuro). This is the largest decision in game. If you follow the Iron code, a boss fight will immediately start followed immediately by another. Success takes you directly to the Shura ending, and the game immediately ends. This is the 'Bad' ending and also quickest time to completion.

Choosing to betray the Iron code also immediately starts a boss fight, however, upon victory the game will continue. Sekiro can now use the items gathered at an incense burner, travel back to the location where he received a macguffin, and gain access to the last environment in game, Fountainhead palace. Here... things get tricky.

Immortal Severance

The Immortal Severance ending will be earned by just following through the rest of the game with no extras. Just beat the boss at the end of Fountainhead and follow Kuro immediately after returning to Ashina.

For both the Return ending and the Purification ending, side quests are involved that can be started prior to fountainhead palace, but MUST have reached certain stages prior to defeating the final boss of fountainhead palace or they will become locked.

Purification

In Purification, Sekiro must make use of the Eavesdrop mechanic on Lord Kuro while he's in the castle after defeating Owl (Sekiro's dad). This is actually under contention at the moment in the community, but still makes this easier. At some point, Kuro will have a 'vague' statement that alarms Sekiro and Lady Emma (the doctor lady) will change position to the top of the stairwell leading into Kuro's chamber. Speaking to her reveals both are worried Lord Kuro is being too drastic and will die to complete this Severance ritual.

Continue talking and resting at the idol to Emma until she steers you to a few places in game (the old graves and the dilapidated temple). Once she has moved there and is speaking with Sculptor, Sekiro must once again Eavesdrop on their conversation to learn that he or Kuro in fact must die to complete the purification ritual. One must perish. Sekiro takes the responsiblity upon himself and Emma gives him a new bell to be used at the statue of Buddha to take him to an alternate Hirata Estate. Defeating the final boss here grants an Everblossom branch.

Note, this Hirata Estate will override the old estate. It is possible to miss items because of this. Additionally, the new Hirata estate contains 2 prayer beads and a memory, so completetionists must always take this sidequest whether or not they wish use this ending at the end. This will allow for max Vitality and grant 1 additional Attack Power.

Return

Finally for Return Sekiro must complete some really odd tasks for the Divine Child sealed away in the monastery. Once Sekiro has acquired the Mortal Blade, the Divine Child will grant Sekiro rice as long as he has none on him. These can either be eaten himself, or given to the crazy old lady. Giving them to the lady results in clues for completing this quest chain. After a few gifts of rice, the Divine Child becomes weakened and requests a Persimmion. Once given, she will not only grant Sekiro rice, but also a bundle of rice for Lord Kuro. Give this rice to Kuro, and he makes some sweet rice for Sekiro. Eat it, talk with Kuro, and then return to the Divine Child to report this.

After hearing how Lord Kuro is still human, the Divine Child decides to help. I'm fuzzy on whether this had to happen prior to the rice or after, but in any event, a specific tome about the Infested is required that needs to be given to the Divine Child. If Sekiro did not receive it from a priest in Senpou Temple, it can be found in the pond with 2 golden carp and a prayer bead at the bottom. After the child reads from the tome and decides to help Kuro, she moves into the Halls of Illusion where the 4 monkey puzzle fight occurred. She asks for your assistance. Agree, and Sekiro is sent to find 2 'fruits' throughout the world. Spoiler: they're hearts of the Giant Snakes you've been running into. Both fruits involve some odd running around and puppetering some enemies to get you access to locations. In game tips come from the crazy old lady mentioned earlier. Once you have the fruits, give them to the child who will eat them and take on some burden. Return after resting and accessing fountainhead palace. She will now be blind and give Sekiro Frozen Tears along with new, better rice.

To actually get the endings, Sekiro must choose what he gives Lord Kuro upon defeating the final boss. Just giving him the Dragon Tears from Fountainhead grants Immortal Severance. Giving Tears plus the Everblossom branch is the Purification ending. Giving Tears plus the Frozen Tears grants the Return ending. For how those cutscenes play out, I'll refer you to YouTube since this post is massive already :)