I would like to clarify a few things about Samurai Jack. All of the words in TenMinJoe's narration and rap are exactly what I heard. You can tell that Aku does say "My evil is law" because that's true in the future. It just makes sense, and I am quite sure that TenMinJoe is correct about everything he wrote. Also, Aku has an over-the-top ancient Japanese accent because the voice actor is indeed, old and Japanese. :)

"A cross between Bruce Lee and Clint Eastwood. Jack is the strong, silent type - a man of few words... but rest assured he is a noble, nimble, tireless hero." - Los Angeles Times

(The Gist)
In the premiere movie of Samurai Jack (the first three episodes) the story is told in full. Aku is "The Master of Masters, The Deliverer of Darkness, The Shogun of Sorrow!" A magical, shapeshifting wizard who wishes only for control and destruction. Jacks father once imprisoned Aku with a magical sword constructed by three wise men. Aku broke free when Jack was a child, and Jack was sent to train under multiple families, in many countries, to become the most well-rounded samurai. When Jack was ready, he returned and found his mother, who handed the sword down to him. The magical blade bestowed upon Jacks father to destroy Aku, was passed down to attempt to destroy Aku yet again.

Jack returned to his home village, and freed the tortured people (including his father) from the wrath of Aku's minions. Fulfilling his destiny, Jack pushed on to find and vanquish Aku. "Before the final blow was struck" Aku teleported Jack into the future. Aku was never destroyed in the past by the one samurai who could, so his control and power has grown over thousands of years. Aku not only controls all of the Earth, but has branched out and started to grasp other parts of the galaxy as well. Jack seeks to return to his time, where Aku's destruction has not taken place, and stop Aku before his grasp gets too strong.

It's worth pointing out that Jack is not his real name. When he first landed in the future, a few on-lookers simply referred to him as Jack when they were talking about how impressive his skills were. Also, Aku has placed a price on Jacks head. Mercenaries, Bounty Hunters, and anyone after a large amount of money seeks to kill Jack for the reward.

Samurai Jack is not meant to be solely funny, but it certainly is, frequently. I'm not really a fan of anime, and that's not what this is. It was created by Genndy Tartakovsky so it looks like a few of his other creations on Cartoon Network.

(Voices) imdb
Phil LaMarr - Samurai Jack
Makoto Iwamatsu(Mako(I) on imdb) - Aku

One great plus about the series is that it's quite easy to watch any episode at any time, even without any prior knowledge of the show. Samurai Jacks story and goal is spelled out in the intro by Aku, and any reference to past episodes is explained in full.
Here is a run down of the episodes I've seen. I haven't given away anything to really spoil the episodes for anyone, but really... the show can't go on if Jack actually dies or does manage to kill Aku, right?

I, II, III - The premiere movie is 70 minutes long, and so was able to be split up as the first 3 episodes of the Samurai Jack series. In 1, Jacks history is explained in great detail, Aku is introduced of course, and during the fight Aku throws Jack into some sort of swirly teleport that he created. In 2, we realize that Jack has been cast into the future and he is shocked and confused by all the strange people and aliens he meets. In 3, he helps a group of talking dogs defend themselves from Aku.

IV - Samurai Jack accidently gets mixed up with the Woolleys and their cruel, annoying captors the creature-lites(?). The Woolleys were once a great people living in paradise and prosperity, but the the creature-lites rained down, (undoubtedly sent by Aku) in technologically advanced pods with weapons well-built to control the Woolleys. Jack is convinced to free the Woolleys, and after fulfilling this task, their prophetic and wise leader tells him to head north to complete his quest.

V - Jack stumbles upon a group of scientists who have been forced to use their "scientific knowledge to build Aku's robot army". They're attempting to escape from Aku's clutches in a rocket they've built, but they need some way to fight Aku's border patrols around the planet. The plan is to have Jack fight off the inevitable attack, and as long as he is in the escape pod at the back of the rocket, as they approach light-speed they will jetison the pod, which will teleport Jack into the past.

VI - In this episode, Jack finds the shopkeeper whom the Woolley leader told him about. The shopkeeper tells Jack of a jewel in the desert to the West, with tremendous power. Before Jack leaves, a girl teams up with him to help him find the jewel. She seeks to destroy Aku as well, for enslaving her father. Watching this episode more than once is a good idea, there were many hints to the twist at the end. A personal favorite of mine.

VII - Jack learns of a tower guarded by three magical archers, able to defeat impossibly large armies. Anyone able to live to make it to the top of the tower, is granted any one wish by the well. Jack has to figure out why the archers are so good, and use this, with the help of his training with monks in his younger days, against them. Another favorite, but I have a thing for monks.

VIII - Jack learns that there is a reward, one googleplex, for his capture, dead or alive. He bests a large number of Bounty Hunters sent to collect the reward, while Aku says to himself "Is there no fighting style that can defeat his? ... Yes... No fighting style CAN defeat his!!!". Aku cleverly creates a clone of Samurai Jack, to defeat him.

IX - Samurai Jack hears a tale of the fish/people The Triseraquin, caught and released by an old fisherman, and in return for the fishermans releasing of him, the Triseraquin told him of Oceanis, the city under the sea, and the time machine kept there. Jack finds Oceanis and is welcomed warmly, for they have not had visitors in many years, since they were banished to the bottom of the ocean by Aku.

X - An ancient warrior seeking revenge against Aku for destroying his village was imprisoned in a crystal, never allowed a warriors death by the hand of Aku. Over time the warrior learned to control the rocks he was buried in, and constructed a mountain of traps around him, hoping one day to be slain by one brave and skilled enough to pass. Samurai Jack hears his voice calling from a distance and finds him.

XI - Crossing an insanely long rope bridge, Jack is confronted by a rather large Scotsman going the other direction. Neither of them wants to risk falling off the bridge for a perfect stranger, so neither will step aside. after a lengthy battle on the bridge with neither side winning, Bounty Hunters show up, and it's not clear which person they're looking for, because both Samurai Jack and the Scotsman are wanted by Aku.

XII - Samurai Jack joins a gang of mobsters to steal the Neptune Jewel. Jack defeats the three guardians of the jewel (which Aku has not been able to do) to give it to the group, who plan to give it to Aku. Jack knows all this and is going along with it because they will lead him to Aku. This episode has a lot of funny Reservoir Dogs references, with Mr. Blonde, Mr. Pink, and Nice Guy Eddie.

XIII - Bottom line: Best Episode EVER. The tales of Samurai Jack's frequent (and almost always successful) battles with Aku have spread, and children are acting out the battles, mocking and laughing at Aku. Aku attempts to convince the children to love him, so that they won't rebel against him as they grow. They would rather hear stories about Samurai Jack though, so Aku puts some twists on some classic fairy tales to make Jack look evil.

Start of Season Two XIV - Jack runs into a tribe of monkey-like creatures. While under their wing, Samurai Jack witnesses them being attacked and overrun by another group, stealing their food. Jack teaches them how to defend themselves from the rival tribe of thieving gorillas, and in return, they teach him how to "jump good".

XV - Samurai Jack meets a magical wyrm that can supposedly grant wishes, then a family of metal-eating robots dressed as poor helpless people, then a fairy held prisoner by a gargoyle. This episode makes no sense and doesn't explain anything. I think it was a dream

XVI - Jack is captured by a group of robots and taken to the "Dome of Doom" where he's forced to participate in Gladiator like fights to the death.

XVII - The Scotsman from episode 11 returns, and tells Samurai Jack of his wife, who's been kidnapped by the Master of the Hunt. In order to free her, he is only allowed the help of a stranger, so he calls upon the aid of Jack.

XVIII - A scientist builds 8 robots of Adamantium specifically designed to destroy Samurai Jack, in an agreement with Aku. Aku, as usual, lies and destroys the scientists home city. Because of Aku's betrayal, the scientist helps Jack destroy them with a bionic arm of sorts. Features divine intervention!

XIX - Samurai Jack finds himself in the empty, moss-covered remains of his home village. Of course this brings back memories of his life as a child, and we get to see a bit of background on why Samurai Jack is so helpful to everyone he comes across. I'm told this episode has Lone Wolf and Cub references in it. I haven't seen the movies, and the episode was still so smooth that I didn't notice anything out of place, so that Genndy is clever.

XX - Jack follows 3 monks and climbs a mountain. "It is impossible" is uttered more than once. Mostly though this episode serves as a motivator for Jack's goal, despite the 20 let-downs so far.