How does Haki works? Is it same as Chakra, ki and Nen?

Haki is a manifestation of inner strength/willpower/spirit. It is similar to chakra/ki/Nen.

There is no implicit mystery behind it or rather I did not pick up on hints that there is a secret. It is a physical/spiritual law of One Piece universe that does not really exist in our one. Only Piece characters can use haki.

Haki can become stronger, especially when a Haki user pushes themselves; and their Haki to the limit and beyond in battles against far stronger opponents.

Haki exponentially grows whenever a user fights a more powerful opponent. Throughout One Piece, Luffy has overcome major obstacles like Donquixote Doflamingo and Charlotte Katakuri. His endless determination is what allowed him to prevail in the end.

There are three types of Haki and they can be read on said wiki:

Observation Haki/Mantra allows the user to detect the spirit of other people. It allows him to detect where they are and predict any attack they are about the throw.

Armament Haki which hardens a weapon/skin/clothing of the user at will to protect against blows. This allows the user to hit with his spirit and hurt logia users that are immune to normal blows with their fluid state.

Conqueror's Haki allows the user to force his will at a distance towards others. This can induce terror in the recipient or cause them to pass out. Often this is used to convince beasts to flee or follow the orders of the user. Apparently it can also be used to push away inanimate objects as was done by Shanks on Whitebeard's ship.