Is there a definitive 'Best Weapon' for each weapon type in Skyrim?
So I was wandering up to the forge in Whiterun with my Ebony Ingots, Daedra Heart and other assorted bits and pieces in hand. I started to go through the process of smithing out my lovely new one-handed Daedric sword, but then I got thinking; is there even any point to me making this? Is there already a better weapon out there that will have more damage (even after I improve it with my 100 smithing, smithing improvement potions and my smithing outfit with Fortify Smithing enchantments).
With all the available enchantments, ability to improve weapons via smithing, and skill improvement potions, is there still a definitive list of 'best' weapons in the game?
This answer refers to equipment available in Vanilla Skyrim, without any DLC. I've been informed that there are some notable new items in Dawnguard that warrant inclusion in this list, but until I can play it, I won't be updating this answer.
To pile onto the existing answers, the only time you wouldn't want to use a Daedric weapon is if you're interested in investing fewer perks into the Smithing Tree, while still investing more than 0.
Essentially, at 0 perks, Daedric retains its advantage over a weapon of any other material once improved, (with the notable exclusion of the two noted exceptions in terms of base damage, the Blade of Woe and Chillrend.)
With the Daedric Smithing perk (and all of its pre-requisites), a Daedric weapon will again trump everything, (including in this case, the exceptions like the Blade of Woe, which do not benefit from the Smithing perks).
But, if you're moving up the Light Armor side of the Smithing tree, or if you prefer to spend fewer perks, you may find upgraded Glass, Elven, or even Steel equipment outperforms Daedric thanks to the enhanced upgrades. This is particularly noticeable at very high Smithing levels such as those you'd achieve by stacking Fortify Smithing enchants and potions - particularly if you do so recursively.
Finally, and to the real meat of your question: Are there any unique weapons whose enchantments make them a better choice than a custom enchanted (or Unenchanted, Elemental Fury'd) Daedric _______?
And the answer to that is, it depends.
For raw damage output, as has been noted, Daedric will generally win, with the above noted edge cases providing notable exceptions. However, there are a few unique artifacts that are worth keeping in mind - especially if you have a high enough enchanting skill to stack Fortify 1/2 handed enchantments to a point where base damage becomes somewhat irrelevant. The reason these are useful generally has less to do with their raw power, and more to do with the unique utility they provide.
The Ebony Blade - There are a few things that are notable about the Ebony Blade, but the most important is that its Absorb Health enchant is chargeless, meaning it has unlimited uses, and that when used against an NPC who 'trusts' you (i.e. a follower, or someone for whom you have completed a quest), the attack is not considered an assault. This makes the Ebony Blade an ideal weapon for assassinations, or for merely killing shopkeepers who looked at you funny. The Ebony Blade also has some interesting interactions with both your one and two-handed weapon skills that you can read about here.
Dawnbreaker - When this weapon is used to strike the killing blow against an undead foe, it causes AoE damage to all other nearby undead, with a chance to cause them to flee. This can be handy in particularly dense crypts.
Mehrunes Razor - has a chance to instantly kill your target on hit. Combined with the Razor's fairly high base damage, this is probably the best dagger available for pure lethality without Smithing perks. And competes for the title, even with them.
Windshear - the Knockdown effect provided by this weapon is unique. There is no other way to obtain it.
Dragonbane - Does a large amount of additional damage to Dragons. This additional damage can be a fairly substantial increase from what's achievable with other weapons early in the game, but with higher level smithing, is eventually smaller than the gain by upgrading to a higher quality blade.
Most other unique weapons - artifacts included - provide effects that you can duplicate with the enchanting bench. As a result, while some are useful, they will generally be outperformed by crafted weapons if you're willing to take the Smithing perks, and in many cases, even without them.
A Note on Dual Wielding: There's one other thing that should be noted with regard to weapon selection that hasn't been noted in any of the answers here, specific to dual wielding. As yx. notes in his answer, Maces offer the highest base damage per swing, and the most potent weapon specific perk. Daggers offer the fastest swing speed. What's interesting is that for dual-weapon attacks, the speed is based on the speed of the weapon in your offhand. This means that with a Mace in the MH and a Dagger in the OH, you can swing with a mace at dagger speeds. In short, when dual wielding, all things being equal, the optimal combination of weapons is a Mace MH Dagger OH.
A Note on Bows, Maces, Etc.: No, the above list doesn't include any Bows. It also doesn't include any Maces. Or Warhammers. The reason is pretty simple: There aren't any worth noting. All of the 'unique' bows, maces, and other weapon types not mentioned here are not actually unique in their specific mechanical properties. You could exactly duplicate or improve upon all of them with existing in game crafting mechanics. The thing that sets apart the blades mentioned above is that they all possess properties which you cannot acquire by any other means. For any other weapon type, you should compare the best results you're capable of achieving with your current enchanting skill to whatever artifact you might be interested in - use the one with bigger numbers. With maxed skills, the crafted and custom enchanted will be the one with the bigger numbers. Without them, your milage will vary as with everything. The only other thing to note is that, in the specific case of bows, there are some unique considerations with respect to weight and speed that will affect your choice.
For base damage, nothing beats Dragonbone♥, point for point and perk for perk.
But honestly, with Smithing, base damage becomes less meaningful.
Consider: A Daedric Sword has only 7 more base damage than the Iron Sword you get in the tutorial. Percentage wise, this is pretty massive -- Daedric is 200% of Iron's damage -- but once you add in the +10 from smithing, you get 17 and 24. Suddenly, Daedric is only 140% of Iron. The higher above Legendary (Smithing: 91) you go, the closer the relative percentages.
This isn't optimal for apples to apples comparisons (Daedric will always be higher damage), but it does show that once you get your smithing high enough, other weapon attributes, like speed, make for a higher effective DPS (Damage per Second) than simply adding more base damage.
So your Daedric Sword will always be the best weapon (highest base damage) by virtue of it being Daedric, but it might eventually start to lose out to a faster weapon in terms of raw DPS (like a Dagger).
[Before Dawnguard, nothing beat Daedric except for daggers, where the Blade of Woe and Mehrunes' Razor beat it out in base damage (1 point) and lethality (chance of instant kill) respectively.
The level 46+ version of Chillrend (a blue Glass Sword) also narrowly edges out daedric, again, by one base damage.]
♥Dawnguard only.
I found that the more I got into the late game, the more I relied on daggers and maces.
Why daggers?
Daggers have the fastest attack speed in the game and the highest sneak attack multiplier (30x). As Raven points out, when you have the maximum smithing capability, the raw numbers do not vary much between your weapons and this is when daggers (with an inherent 20% faster attack speed) out shines everything else. As for what enchantments to place on them, the best enchantment is actually no enchantment. Why is that? Because with the Elemental Fury dragon shout, you can boost your attack speed by even more. So I tend to use a Daedric dagger while sneaking around.
Finally, as an added bonus, since daggers are extremely light, power attacks with it cost the fewest stamina.
Why Maces?
Maces have this awesome perk called Bone Breaker, which allows you to ignore up to 75% of the opponent's armor. When you are facing tougher and tougher enemies wearing increasingly better gear, being able to ignore their armor is a huge damage boost when you don't have the chance for a surprise attack. Again, daedric has the highest base damage, and you can further enhance this with the elemental fury perk for faster attack speed.