Useful Weapons
There are numerous weapons which are not artifacts that are still very useful if you are specialized in that variety of weapon. Some are very easy to get early in the game, others are good for enchanting, and others are just good in general.
Daedric Weapons:
Daedric weapons can be obtained in a number of ways:
1: The Easy Daedric Weapon Quest: This handy quest allows you to claim a daedric weapon of whatever weapon skill you want. Go to St Olm's Canton, and go into the haunted manor house. Downstairs there should be a character called Dram Bero, who will tell you about the Vassir Didanat mine. Go to the mine by crossing the wooden bridge South of Balmora (the one with the egg mine nearby). Keep going in that direction until you hit the mountain wall. Search around there for the mine entrance. Go inside the mine, then go back to Dram, and he will give you a choice of weapons.
2: The Dremora Murdering Technique: Once you are strong enough to take on a Dremora Lord (probably about level twenty, although this will vary a lot), you can easily acquire Daedric weapons. Head down to a shrine, and take the offerings from the altar. A Dremora Lord will teleport in and attack you. Kill it and take its weapon. Take the offering to an open area, and drop it on the ground. Now pick it up again, and a new lord will be summoned. You can repeat this process as many times as you want until you get the desired weapon. Failing this, create a summon dremora, or summon dremora lord, spell, and cast it. Murder the summoned daemon, and just as it dies, loot the corpse. You should have a split second to access its inventory before the body disappears, allowing you to summon up daedric weapons. Note that these methods will never net you a bow.
3: Set Locations: These items are always found in the following locations. I have left out weak ones.
Battle Axe: Ibar-Dad, daedric ruin W of Urshilaku camp. Also with several other daedric items.
Claymore: Telvanni House vaults.
On a pedestal, Mausur Caverns, NW of Dagoth Ur, Red Mountain
Gathal Llethri, Guard House, Dren Plantation. Kill or steal.
Ulms Drathen, Armigers Stronghold, Molag Mar. Kill or steal.
Dai-Katana: On a corpse, Maesa-Shammus Egg Mine, E side of top of Mt. Assarnibibi.
Llaro Llethri, Tower Shack, Dren Plantation. Kill or steal.
Bolvyn Venim, Venim Manor Private Quarters, Ald-ruhn. Kill or steal.
Katana: Galos Farethi, Dren's Villa, Dren Plantation. Kill or steal.
Longbow: Forgotten Galleries, Maelkashishi, daedric shrine W of Maar Gan, over the mountains.
Longsword: Relas Arothan, Assernerairan, daedric shrine Vivec St Olm's Underworks.
Arver Rethul, outdoors at the Dren Plantation. Kill or steal.
Mace: Sunken Vaults, Ashalmawia, daedric shrine just E of Ald Velothi.
Shortsword: Aryon, Aryon's Chambers, Tel Vos. Kill or steal.
Gentleman Jim Stacey, Lower Waistwork's bookshop Vivec Foreign Quarter.
On a dead body in Dagoth Ur Lower Facility.
Spear: Underground section of Berandas, Dunmer stronghold.
Staff: Ibar-Dad, with Battle Axe
Kora-Dur, E of Kogoruhn, unnamed Dunmer Stronghold.
Wakizashi: Dead body, Endusal, E of Ald-Ruhn.
War Axe: On Gordal, Ashalmawia, same section as Daedric mace. Also has almost full ebony.
Ammunition: Daedric arrow: Top level of Berandas stronghold, and by the daedric bow.
Ebony arrow: Indarys Manor, Bal Isra (on the paper map, West of the Ghostfence). Also Rethan Manor, Odai Plateau (on paper map, S of Balmora). These are the Redoran and Hlaalu strongholds.
Daedric Dart: By the daedric mace. Severa Magia, Ald Sotha, Lower Level. Kefka burial, Urshilaku burial.
Ebony Dart: Rethan Manor, Redoran Vaults and Suran Pawnbroker.
Ebony Throwing Star: Ahnassi's last quest, Indarys Manor, Galom Daeus Entry area.
Bonemold Longbows: Useful, since they have the second highest enchant capability in the game.
Found on various archer NPCs, random drops, and the Savel Ancestral Tomb, on an island directly E of Molag Mar. Tel Aruhn Smithy, Tel Vos Barracks and Armory.
Dwarven Pneuma Trap: Sort of unique, but not that good. Dwarven Battle Axe with soul trap on strike. Found in the locked room of the Razor Hole in Balmora.
Dwarven Crossbow: The second best marksman weapon, with a set damage of 30.
Found in random Dwarven chests, on a guard called Cun outside Tel Branora (you get a quest to kill the group which this person is a part of if you talk to people inside Tel Branora). Also, Llivas Othraven in the Ebonheart Grand Council Chambers. Kill or steal.
Ebony Staves: The highest enchant value of any weapon, a huge 90 points, enough for several very powerful enchantments. Found: Ghostgate Tower of Dusk, Lower Level; two in houses at Tel Uvirith; on Goris the Maggot King in the Venim Ancestral Tomb, N of the Zainab Camp. Tholer Saryoni, High Fane, Vivec Temple; and Ernse Llervu, Abbey of St Delyn, Vivec St Delyn's.
Ebony Shortswords: Bizarrely, have a better weapon than a daedric shortsword: Daedric weighs 24, does 10-26 damage, and ebony weighs 16, and does 15-25 damage. Better off still with a daedric wakizashi.
Ebony shortswords are found: Rumare, outside Molag Mar; outside on Odai Plateau (by the Hlaalu stronghold); Dhaunayne Aundae in Ashmelech, island E of Valenvaryon; a chest in the hut labelled Verethi and Dralor in the Dren Plantation; Upper Level of Druscashti, NW of Falasmaryon; Galas Drenim, Ebonheart Grand Council Chambers (kill/steal); Tower of Dusk Lower Level, Ghostgate; on a body, Hall of Maki, Kogoruhn (unlabelled Dunmer Stronghold); Aerin, the Maar Gan tradehouse; Gols's house, Rethan Manor; Procyon Nigilius, Sadrith Mora Mages' Guild; Arena Underworks and Redoran Vaults, Vivec; and finally, Salver Lleran, Telvanni Sorcerer in their canton, Vivec. Thanks to the UESP, there, or I'd have been in the editor for far longer than necessary.
Glass Claymore: A strong but light two handed sword. Looks great as well.
Tuveso Beleth, Ald-Ruhn Smith, and Cania Mico, Tel Vos Barracks and Armory.
Glass Halberd: One of the better spear weapons.
Underground, Bal Ur, shrine N of Suran; Hall of Maki, Korogruhn.
Jinkblades: Very useful short blades with paralyse on strike. Wonderful for a stealthy hit, followed by repeated attacks while they are still paralyzed. Three types: steel, Dwemer, and the semi-unique Steel Jinkblade of the Aegis. Steel can be found all over the place, on NPCs and at random.
Dwemer can be bought from the Balmora Fighters' Guild Smith, and found in the Tel Vos Service Tower and in the Forge of Hilbongard, Forgotten Vaults of Anudnabia, Omaren Ancestral Tomb, on an island E of Sadrith Mora. Well worth a visit for treasure.
The Jinkblade of the Aegis only does 4-5 damage, and is found both randomly and on Nileno Dorvayn, in the Hlaalu Council Manor, Balmora. It's paralyse is only for one second, but it also summons a bound cuirass. It works on use, not on strike, and is worth 7000.
There are also numerous glass dagger jinkblades.
Last Rites: A big Dwarven Battle Axe, doing 1-44 damage and dealing 1-25 fire damage on strike.
Found in the NE tower, Tel Vos, and on Birer Indaram, in the Armigers' Stronghold in Molag Mar.
Swords of White Woe: Easy available early on in two locations (Suran Guard Tower and Balmora Eastern (I think - if not, check the others) Guard Tower). They have to be stolen using invisibility or chameleon, but they are ebony broadswords (4-26 damage) with a drain health enchantment worth 17000. Great for early players.
Phew, at last. This bit's taken me ages.