Some characters in D&D don't really care about weapons, Especially given the existence of Cantrips, a Wizard or Sorcerer isn't going to be really that concerned with whether they have a quarterstaff or a dagger.
But if you're one of the many martial classes in D&D, weapon choice will make a difference.
Classes come with their own weapon proficiencies. Some get all Simple Weapons, some get both Simple and Martial weapons (which means everything except improvised weapons) and some get a more curated list - like Rogues, who typically stick with simple weapons but can also use things like Rapiers.
Weapon choice can have a big impact on the kind of damage you're pouring out. So what weapon should you focus on?
Naturally, if you come across a super magic Pike that does an extra 3d8 psychic damage or something like that, you can probably choose to go with it even if you've typically been using a Greatsword, but when dealing with mundane items, there are considerations to be made:
Many weapons are one-handed, which means what it says - you can hold it in one hand. This leaves the off-hand open for a shield, to cast spells with, or to hold another weapon in the off hand.
Some one-handed weapons have finesse, which allows you to use Dexterity for attack and damage rolls, which is otherwise only used for ranged weapons. Note that a Monk will effectively turn the weapons that count as Monk weapons into finesse weapons even if they aren't - so you can use that quarterstaff with dexterity.
One-handed weapons cap out with 1d8 damage - which you'll get with Longswords, Battleaxes, Warhammers, Rapiers, and Flails (and I think maybe Morning Stars?) If you can take one of these, it's typically your best option for a one-handed weapon.
However, if you want to dual-wield, fighting with a weapon in each hand, you need to get weapons with the Light property. The highest damage these typically do is 1d6, including shortswords and handaxes. Absent the two-weapon-fighting fighting style, you don't get to add your modifier to the damage off the off-hand strike. Still, the existence of magic weapons and other on-hit effects does mean that you can wind up cranking out more damage this way, so it's a viable option even if you don't take this fighting style. The other big catch with dual-wielding is that your off-hand strike will take up your bonus action, so if you are a class that likes to use those frequently, this can gum up your personal action economy. Note that the Dual Wielder feat allows you to dual-wield non-light weapons, so if you take this, you can fight with a pair of battle-axes or warhammers.
Some weapons, including Longswords, Battleaxes, and Warhammers, are "Versatile," which means that you can switch to a two-handed grip and do a bit more damage with them, typically increasing the damage to the next higher die (these go to 1d10 from 1d8). This is a good option if you want the chance to pump out a little more damage, but I recommend that you instead commit to two-handed weapons if you want to really focus on that damage.
There are a fair number of two-handed weapons, and they come in a couple of categories:
Pikes, Glaives, Halberds, Staves and Spears all count as Polearms for the sake of the Polearm Master feat, and the first three have a special feature called Reach, which extends their melee range to 10 feet rather than 5. The pikes, glaives, and halberds also do 1d10 damage, which makes them equivalent to wielding those versatile weapons, but the added range and potential interaction with Polearm Master make them a solid choice for a weapon.
For the maximum damage output options, you can go with a Greatsword, Maul, or Greataxe, though unlike their one-handed/versatile counterparts, these are not all equivalent in damage. Greatswords and Mauls both do 2d6 damage on a hit, while Greataxes do 1d12.
While these do have a the same maximum damage, the average is slightly higher for the 2d6 weapons, given that the minimum is 2, rather than 1. Also, if you have the Great Weapon Fighting fighting style, you get to re-roll 1s and 2s you roll with the weapon, which will happen more frequently for the 2d6 weapons and thus generally boost your average damage more. (2d6 winds up giving you about 8 and a third average damage with this fighting style, while 1d12 gets 7 and a third). So why would you ever take the Greataxe? Well, first off, you might just want to be able to roll that big chunky d12. But if you're playing a half-orc or Barbarian (or a half-orc barbarian,) you'll get to roll additional dice, with more as you level up, when you get critical hits, and adding a d12 rather than a d6 to your crit is obviously better. That being said, if you do some somewhat complex math, the total boost to damage here only kicks in once you're rolling a lot of extra d12s (meaning you'll have to be a very high level Barbarian) but it will, eventually, offset the lower damage for ordinary hits.
Still, this is super min-maxy, which means you should just go with what you want.
Now, mechanically, when does it not really matter what weapon you take? When is flavor really going to be the primary consideration?
Monks get to use their martial arts die in place of the weapon's typical die if they choose. While a Monk will be better off using their quarterstaff versatile-style (for 1d8 damage in two hands), once you hit tier 3, your martial arts die will already be a d8, meaning anything you fight with (including fists) will be doing that anyway.
Rogues will still use their weapon's typical damage die, but the vast majority of the Rogue's damage comes instead from Sneak Attack. By level 5, you're dealing 3d6 additional damage when you get your sneak attack in, which is roughly 10 damage. Comparing a dagger's average damage (about 2.5) with a Rapier's (4.5) means a difference between about 12.5 and 14.5 damage, which means only a 14% loss in average damage. Once at higher levels, say level 11, when your sneak attack is 6d6, or about 20 damage, that goes down to a 9% damage loss. (A rogue also might consider preferring light weapons, as your off-hand damage can also trigger sneak attack if you missed with the first attack, though this will mean not being able to use your bonus action to dodge, disengage, or hide).
Some oddball weapons can actually be very flavorful, even if they are strictly worse than others. For example, a Flail, like a Warhammer, does 1d8 bludgeoning damage, but it doesn't have the versatile property. Still, if you're committed to fighting with a shield, this winds up not making any difference, and flails are pretty cool (and historically very hard to fight effectively with!)
In fact, for DMs, I recommend giving the party the weirder weapons as magical items to force the party to consider the funkier options. Take the Lance, for example, which is a one-handed weapon that deals 1d12 damage, and has reach, but has the very odd downside of giving you disadvantage on attacks if the target is within 5 feet. I've literally never seen anyone ever use this weapon (even NPCs) but it could be really fun to make a powerfully magical one that forces your players to consider using it even though it has one very big downside.
I have a vampire-themed adventure that I'm theoretically in the middle of with my original campaign, though I've got a whole other campaign I'm running now, and in it, there's a Castlevania-inspired magic whip that deals additional damage to the Undead - compensating for the whip's typically low d4 of damage.
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