I wonder why they needed the Death Star as I’ve seen in comics and YouTube videos that a few Star Destroyers can basically destroy a planet, though not to the extent of what the Death Star did to Alderaan. Isn’t that overkill?
In Darth Vader, the 2017 Marvel comic, Tarkin almost destroys Mol Cala with his Star Destroyers.
Grand Moff Tarkin gained his favour with the Emperor in large part due to his belief that the threat of destruction was an effective tool for keeping rebellious systems in line.
As he put it "Fear will keep the local systems in line, fear of this battle-station"
Destroying Alderaan was meant to be the act that demonstrated to the galaxy exactly what extremes the Death Star and the Empire were capable and willing to do if you stood up to them, so that nobody would be willing to stand up and be turned to space-dust.
In the strictest sense, the Empire didn't need the Death Star, it was simply the biggest and loudest expression of how they did things.
Prior to the Death Star, the Star Destroyer held this role. With a single star destroyer overhead (visible from the surface of a planet to the naked eye) being more than enough to quell rebellious worlds.
Imagine seeing the huge rising globe of the Death Star coming up above the horizon and eclipsing the sun, its great Eye-like superlaser dish aimed your way..
The ultimate Sword of Damocles over your head.
It'd be a rare being who would stand and face that.
The Death Star is an unassailable platform for warfare
Aside from the Tarkin Doctrine, the Death Star has also been described as an Expeditionary Battle-Moon. It's not just a weapon, it's a mobile fortress.
Against a fleet of warships, it presents an extremely robust target, mounting planetary-scale defences (which easily shrug off most capital-scale weapons) and huge surface-armaments to kill anything small enough to get close.
Quite aside from that, the hull and superstructure is kilometers deep before you get to any critical systems. It would take a sustained and focused bombardment through kilometers of armour and interior spaces to disable the Death Star, and if it felt remotely threatened by that it could hyperdrive away.
Fighters were assumed to be utterly irrelevant to something that large. An elephant doesn't care about mosquitos except to swat them in annoyance.
On top of its general robustness, the Death Star carried millions of troops and their support-vehicles and landing craft. It could easily support an invasion-army to quell a rebellious planet the old-fashioned way if necessary.
It had dock-space onboard for ships of all sizes up to and including its own escorting Star Destroyers as well, allowing it to serve as a mobile starport and supply facility for a fleet-action.
The Death Star is a toolbox for its commander to accomplish pretty much anything the Empire demands.
It represents a new paradigm of warfare based around utter domination of the battle-space. It has no peers. There are no Rebel Deathstars or planetoid-sized warships to face one-on-one. (At least, not until decades later in Legends-Canon when the Yuuzhan Vong show up with their worldships..)
I think the answer is "shock and awe" - the Empire could send ships that can destroy a world's ability to sustain life, but that's not as impressive as destruction of a whole planet down to atoms in mere moments by a single massive vessel. The Death Star is intended to terrify.
HAN: Our position is correct, except...no, Alderaan!
LUKE: What do you mean? Where is it?
HAN: Thats what I'm trying to tell you, kid. It ain't there. It's been totally blown away.
LUKE: What? How?
Ben moves into the cockpit behind Luke as the ship begins to settle down.BEN: Destroyed...by the Empire!
HAN: The entire starfleet couldn't destroy the whole planet. It'd take a thousand ships with more fire power than I've...
When the Death Star was first encountered, the Imperial Navy was not believed capable of destroying a planet with its existing materiel.
General Tagge : That's impossible! How will the Emperor maintain control without the bureaucracy?
Grand Moff Tarkin : The regional governors now have direct control over their territories. Fear will keep the local systems in line. Fear of this battle station.
This is the basis of the Tarkin Doctrine. Basically the Death Star is meant to be seen as undefeatable and unavoidable so any system considering dissenting against the Emperor would rather do anything than be on the wrong end of its primary weapon.
In practical terms, it is only one station and due to its huge size, it takes ages to actually get anywhere (relatively speaking anyway) but that is not the point. The point is any system would do pretty much anything and agree to pretty much anything to ensure it doesn't show up in orbit around their planet.
It's a deterrent.
The point is to deter rebellion, i.e. to make others not even think about trying, and deterrence isn't just about simply being materially effective: it's about being psychologically effective, and that usually requires adding something above and beyond what is strictly practically necessary from the viewpoint of simply, say, terminating a rebellion's activity, so that the mere risk of it is too scary to contemplate.
A fleet of ships pommeling your planet gives at least the sense - and maybe even a little of it may be justified - you can escape or at least resist.
But a "big bad beam 'o doom" that comes out of nowhere and instantly sends the whole planet up in an explosion as powerful as a nova, generally is more chilling because of its instantaneous and thus leaves-no-hope feel.
Planetary shields and orbital battle stations are a thing in the Star Wars universe, even a common thing -- just looking at the films, we see the shield around Scarif in Rogue One, the shield surrounding the moon of Endor (which the Rebels only got through by having an old security code), the one around Starkiller Base (which Han got through with some fancy hyperspace tricks), and the smaller theater-shield over the base on Hoth (which could nonetheless "deflect any bombardment" and forced a ground assault by AT-ATs). While some of those could theoretically be battered down with enough firepower, it would potentially require a whole fleet of Star Destroyers and a great deal of time.
The Death Star renders planetary defenses totally moot. If they fire the big gun, it doesn't matter what shields are in place, and the beam can reach from far beyond low orbit (as seen in the destruction of Alderaan), beyond the range of even large defense platforms. Where a well prepared world might be able to repel a whole fleet indefinitely, the Death Star's superlaser could casually destroy the planet regardless.
Even if the Imperial fleet could eventually break any planet, which is not clear, an extended siege is a bad look for the Empire. It's not just resource-intensive; it gives the friends of the besieged time to spread the word, to turn public opinion against the empire, and make the besieged into martyrs. There's a reason we still remember the Alamo, the Masada, Troy, and so on. The Death Star denies the romance of a force holding out against a superior enemy. It just wipes them away along with the entire planet that potentially supported them. It's a terror weapon that makes it everyone's job to keep rebellion off their planet because the alternative might mean everyone dies together. Ultimately, it's a way of coercing the populace into becoming collaborators because the alternative is too frightening.
The Death Star is an intimidating display of destructive power, technological capability and resources from the Empire. As such it serves a psychological, military and political purpose.
The Death Star is however not needed by the Empire, but by its rulers.
Indeed, being mobile and hard to destroy, it is the best place for the Empire rulers and generals to govern and oversee military campaigns, surrounded by their most loyal servants, able to stay close to the action while being protected from most threats that could hit them.
A land-based palace might indeed be a target for bombing or terrorist attacks, while even a Star Destroyer can be destroyed in a battle. The Death Star is comparatively a safe lair for the Emperor and Darth Vader.
From a more out-of-universe, symbolic and interpretative point of view, the Death Star is an expression of the Empire’s ideology: that of a totalitarian autocracy. One of my favourite bloggers, in a post dissecting how the Death Star is portrayed in Rogue One (some spoilers in the link), argues:
[L]et's be clear: the Death Star is the Empire.
In the OT we don't ever see Coruscant, or any Imperial center of power besides this looming metal moon. It is what we know of the Empire (particularly in ROTJ, when it becomes the Emperor's throne room and where he is defeated.)
As such, it follows the logic of the Empire, which is centralization. It's cruel and inhuman to lash every sentient being in the galaxy to one ruler. It's also not anywhere near efficient or reliable. The desire to do this is not a rational desire, but an ideological one.
The instinct that says "Why do we need all these different opinions going around, it's just messy" is also going to say "Why do we need all these Star Destroyer groups around, they could split off or get confused." Better to centralize.
This ties in with themes later explicitly established in the prequels, where Anakin switching his loyalties to the Sith is driven, among other things, by his frustrations and disillusionment with the pluralistic democracy of the Republic:
ANAKIN: I don’t think the system works.
PADMÉ: How would you have it work?
ANAKIN: We need a system where politicians sit down and discuss the problem, agree what’s in the best interest of all the people, and then do it.
PADMÉ: That's exactly what we do. The trouble is that people don't always agree.
ANAKIN: Well, then they should be be made to.
[…]
PADMÉ: Sounds an awful lot like a dictatorship to me.
ANAKIN: Well, if it works…
— Star Wars Episode II: Attack of the Clones
The Tarkin Doctrine mentioned in Ruadhan2300’s answer makes this ideology just as much explicit in the original trilogy.
