There is a lot to say about artifacts, but I will try to keep it simple. This article will summarize the different types of artifacts and how they are acquired.
Doom - Stolen Artifacts
Inside the dungeon doom spawn some "locked" artifacts. They stick to their position until you use the Stealing skill on them (hence the "stolen"). This is a profitable use of your thief character. Some of these are weapons but the main part are decorating items.
The most sought after artifact are the Inquisitor's Resolution gloves which are worth millions.
Doom - Gauntlet Artifacts
The gauntlet in the heart of the dungeon Doom is where spawn the initial artifacts. Those were the first artifacts to appear in Ultima. They are very powerful items and are usually worth a whole lot of gold (from 1M to 30M+). The chance to drop an artifact in the Gauntlet was initially based on pure dumb luck. (Well, the reason why I never bothered *grins*) But at the end of 2007, the system was changed to match the Treasures of Tokuno system. Now your chance to get an artifact increases each time you kill a dark father.
For a complete list of artifacts and their properties, I recommend you go and read either UOGuide or Stratics (see the link section).
Often referred as "marties" the minor artifacts are named items which good properties though not insanely powerful. You can get them through four means. Either, killing mini-bosses in the new Mondain Legacy dungeons, killing paragons in Ilshenar, digging treasure chests or killing Leviathans in the seas of Britannia.
Mini-bosses are named monsters you can find in the new Mondain Legacy dungeon. Each time you kill one (the swoop and the miasma being farmer's favorite) you have a chance to get a minor artifact.
The paragons are powerful monsters spawning in Ilshenar (see the hunting section) and they sometimes have a "paragon chest" as loot. This chest sometimes have a minor artifact inside.
The treasure chests are found using a treasure map. You can find those as loot for killing monsters. Only the highest level of treasure chests does spawn minor artifacts.
The Leviathans are sea monsters you can spawn using a fabled fishing net. Those nets are found in the Ancient SOS treasure chests (see the hunting section about MIBs). When you kill a leviathan, you have a chance to receive a minor artifact.
Note that some minor artifacts can only be received through one of those 3 methods (seahorses from Leviathan, etc.) so you'll need to hunt the particular enemy to have them.
Treasures of Tokuno
The treasures of Tokuno (ToT) were events taking place in the Tokuno Islands. They used to run for a period of 2-3 months. The principle was simple, you killed Tokuno monsters and for each kill increased your chance (proportionally to fame) of dropping a minor tokuno artifact.
In the second stage of the event, you could turn in 10 minor artifacts for 1 major. As this event was limited in time, the price of the items will go raising with time too.
I want to bring your attention to this very particular major Tokuno artifact. Those dyes are the only way to dye metal armor and artifacts! They are limited in charges, so their prices are going to raise insanely with time. In fact those dyes can dye anything that is made of special materials. This included all special leather, ingot and wood types.
In the first ToT event there was a minor "bleach" pigment. It allows to remove the color of an item and return it to its default (iron for metal, brown leather, etc...). A quest was introduced that allows you to get such bleach pigment as a reward. Check UOGuide for details.
The Peerless bosses were introduced with the Mondain Legacy expansion. There is an article about Peerless in the Hunting section so I won't go into details. You just need to know that some items can only be acquired through killing peerless AND dumb luck. You can do peerless for a year and don't receive anything good.
Most of those items are named after the Peerless boss that spawns them (mark of travesty, scepter of the chief, etc.) and there is one that spawns on all, the Crimson cincture.
Most of those artifacts are godly and worth a lot.
Introduced in 2007 with the 10th anniversary event, those artifacts spawn in the Anti-Virtue dungeons (shame, despise, etc.) and are acquired through a method similar to the ToT system. Those items are rather powerful, though you cannot repair them (except for the virtue armor parts which has Self Repair).
With Mondain Legacy, you can acquire item recipes in Heartwood through the crafter quests. Some of those recipes can give you named powerful artifacts. Though, in order to create those items, you need to have many recipe ingredients which can only be acquired through killing Peerless bosses (there is an article about them in the Hunting section).