Haki is one of the two major power types in the One Piece world. While Devil Fruits were the very first power system to be introduced in the series, Haki followed after and has, since then, become ever-present. Haki was first introduced by Shanks in the very first chapter of the story when he used it to scare off the Lord of the Coast, however, its usage was only explained to the fans properly once the story was heading closer to the end of the first half of the series.
When Rayleigh decided to train Luffy, Haki was properly explained as a whole new power that dwells within everyone in the One Piece world, yet, only a handful manage to awaken it due to the various complexities of this ability. Haki and its many types have become very prevalent in the New World, but fans often wonder where it ranks in comparison to some other power systems, such as Devil Fruits, or if it's even necessary for fights, to begin with.
Haki Before the Timeskip
To properly understand the importance of Haki in the story, fans simply need to look at the time when none of the Straw Hats possessed a mastery over this ability. On occasions, Luffy displayed signs of Haki, but he could only control this power after training with Rayleigh. The word "Haki" was thrown around by the likes of Blackbeard, often to measure the willpower of people. When Shanks stepped on Whitebeard's ship, his "willpower" or Haki was praised by Whitebeard's crew. It was quite clear to see that Oda was building up to something with constant mentions of willpower being important to decide the quality of a fighter and, going forward, it became even clearer.
By the time the Straw Hat Pirates reached Sabaody Archipelago, Haki's importance was truly highlighted in the story. The Straw Hat Pirates faced utter destruction on the island, not being able to fight against the Logia powers of Admiral Kizaru or being unable to damage the tough bodies of the many Pacifista on the island. To make things worse, a capable Haki user in Sentomaru showed up and completely overwhelmed Luffy in battle. If anything, it was clear that the Straw Hat Pirates needed to get stronger but, unfortunately, back then, they split up.
For Luffy, however, this was extremely important as he landed on the Island of Women, which is filled with Haki specialists who use what is called the "Kuja Haki." Eventually, on the very same island, Luffy met Silvers Rayleigh following the events of the Paramount War and his discussion with Luffy made it pretty clear that Haki is essential in order to reach the pinnacle of piracy. According to Rayleigh, without Haki, Luffy was just asking to be defeated over and over again. With Haki, however, he'd be a monstrous pirate in battle.
Haki's Importance in the New World
After the timeskip, Luffy, Zoro, and Sanji returned from their respective island having learned the power of Haki. Immediately, fans could see the difference in their level of strength. The same Pacifista that the crew struggled with were now taken down in a single shot, showing how big a difference Haki truly makes. The New World is a sea of elites and only the very strongest manage to survive here. To do so without any Haki users in the crew is simply impossible. In the New World, there are countless Haki users and even Conqueror's Haki isn't rare. It's a power that exists within many in th ocean and, as mentioned by Don Chinjao, only by battling against various other conquerors can one rise to the very top.
Oda didn't make this addition just to make the fighters look fancy with their skills but because it is an essential power in this sea. Without Haki, survival here is impossible. Some abilities simply can't be countered without Haki, such as the many Logias, or even the incredible protection that Luffy has against blunt attacks. While Devil Fruits were certainly introduced first and are, without a doubt, a remarkable power system that allows for some of the creative fights, Haki is just as essential. No power system is greater than the other.
More importantly, Conqueror's Haki, in particular, is a sign of a conqueror and a fighter with incredible potential. Without Conqueror's Haki, nothing would give special attention to those with kingly ambitions that they very much demand. On various occasions in the story, Oda made it clear that Haki isn't the only power and that it can even be broken through via many means. Being a user of Conqueror's Haki isn't the only way to be a top-tier fighter either, as there are many who can reach this level without it. It is possible to reach the top without Conqueror's Haki, however, doing it without any form of Haki is simply impossible.
Haki is a very important power the importance of which shouldn't be underestimated. Regarding the most important goal in the story, finding the One Piece, having Conqueror's Haki is incredibly important, as is having the many forms of this power and to deal with the harsh realities of the New World.