Tall order! Here's a very basic outline of the plot and some key descriptions. MAJOR spoilers, so don't say I didn't warn you!
Star Wars follows the story of the Skywalker family: Anakin Skywalker, his wife Padme, and their twin children, Luke and Leia. It's broken into two trilogies, the first having to do primarily with Anakin and Padme's time, the second, with Luke and Leia's. There's about twenty years between trilogies.
(Episode I: The Phantom Menance)
Anakin Skywalker is a young Tatooine (a desert planet) slave who is found to have incredible Force-sensitivity (the Force is an energy field that grants the trained user many powers. Jedi use the "good" or "light" side of the Force, whereas the Sith and Dark Jedi use the "evil" or "dark" side.) He is taken to the Jedi Temple on the capitol world of Coruscant where he is tested. The Jedi Council (the head of the Jedi) reluctantly agree to train him as a Jedi. At this time, the Jedi learn that the Sith (their ancient enemies) have returned.
(Episode II: Attack of the Clones)
Fast forward ten years. Anakin is now a Jedi Padawan (a learner) under Obi-Wan Kenobi. He meets up with a woman named Padme (whom he first met when he was taken off of Tatooine) while protecting her. To make a long story short, he falls in love with her and vice versa. They marry against the rules of the Jedi Order. At this time, war breaks out (the Clone Wars) between the Republic and the Separatists (a group that is trying to break off from the Republic and overthrow it, led by General Grievous, the Jedi-turned-Sith Count Dooku, and the shadowy Darth Sideous, also a Sith).
(Episode III: Revenge of the Sith)
Fast forward roughly three years. General Grievous makes a raid on Coruscant and captures Chancellor Palpatine (head of the governing body of the entire galaxy). Anakin and Obi-Wan rescue him and return to Coruscant. Anakin learns his wife, Padme, is now pregnant. He begins having prescient dreams of her dying in childbirth and becomes frantic to save her at any cost. A really long story short: he turns to the Dark Side of the Force to try to save her. He commits various atrocities (the destruction of the Jedi Temple among them) and winds up fighting Obi-Wan on the volcanic world of Mustafar. He gets the short end of the duel and is physically injured to the point of needing a life-support suit to survive. Enter Lord Vader (Anakin and Vader are two sides of the same person, Light and Dark respectively). Padme dies of a broken heart (Anakin, in his anger, nearly kills her on Mustafar) but delivers the twins Luke and Leia. Vader is told that the child (they didn't know they were having twins) died with her and begins to serves the newly-appointed Emperor Palpatine (who used to be Chancellor) as his enforcer, earning the fear of the galaxy.
End the first trilogy.
Twenty years pass. The twins have been hidden on different worlds to keep them from Vader and Palpatine. The Jedi Order has been destroyed and the Republic replaced by the Empire.
(Episode IV: A New Hope)
Leia has become a member of the Rebel Alliance, an organization that protests the Empire, sometimes in not-so-peaceful ways. She is carrying the plans to a superweapon called the Death Star when she is captured by Vader. The plans make it to Tatooine where Luke is (and Obi-Wan, under the name of Ben). She is rescued by Luke, Han Solo (a smuggler who gave Luke and Ben passage to where the Death Star was) and Han's first mate, a Wookie by name of Chewbacca. Obi-Wan faces Vader again and dies to let the twins escape (Luke and Leia don't know of their heritage yet). Luke eventually destroys the Death Star while using the Force.
(Episode V: The Empire Strikes Back)
Fast forward a few years. The Empire attacks a Rebel base on Hoth, where Luke and Leia are. They escape and Luke goes to Dagobah, home of the Jedi Master Yoda. He gets about two weeks worth of training before having a vision of his friends being tortured on Bespin. Against Yoda's wishes, he goes to rescue them, but falls into a trap set by Lord Vader. The two duel, Luke loses a hand and gains knowledge of his father (the whole "I am your father" scene between Luke and Vader: it's pretty famous). Han is frozen in carbonite and taken to the Hutts on Tatooine. Luke escapes Bespin with the help of Leia and Chewbacca.
(Episode VI: Return of the Jedi)
Fast forward a few more years. Luke, Leia, Chewbacca and Lando (the one who led them into the trap on Bespin in the first place, now reformed) free Han. A second Death Star is being constructed and so the Rebels plan an attack on the station, located at Endor. Luke returns briefly to Dagobah, where Yoda is dying. He learns of his twin sister Leia from the Force ghost of Obi-Wan, then goes to rejoin his friends. The Rebels go to Endor, meet the Ewoks, and destroy the shield protecting the Death Star. Meanwhile, Luke has told his sister of their heritage and goes to confront Vader, believing his father still has good in him. He is taken before the Emperor, fights Vader, nearly turns to the Dark Side, and finally proclaims himself to be a Jedi after nearly killing Vader. (The Rebels attack the Death Star from space during this confrontation.) The Emperor isn't happy (he wanted to replace Vader with Luke as his apprentice) and so fires Force lightning at Luke to kill him. Luke cries out to his father, who turns back to the Light to save his son, giving his life in the process. Luke escapes the Death Star moments before the Rebels blow it up and meets everyone back on Endor. At the celebration, he sees three Force ghosts: Obi-Wan, Yoda, and his redeemed father, Anakin Skywalker.
Phew! So that's a highly condensed version of the plot of the movies. To specifically answer your questions: in the first trilogy, there's only fighting in the second and third movie (the Trade Federation at Naboo doesn't quite count). There, it's the Republic against the Separtists, led by General Grievous and a former Jedi-turned-Sith named Count Dooku. In the second trilogy, it's Empire against Rebellion.
A Jedi is one who uses the light side of the Force to keep the peace and act as a guardian of the galaxy.
Anakin and Darth Vader are really the same person, 'Vader' being Anakin's Sith name and 'alter-ego,' if you will. Anakin becomes Vader in spirit when he helps the Chancellor/future Emperor kill one of the leading Jedi Knights (Mace Windu) and becomes the iconic representation of evil after his fateful duel with Obi-Wan on Mustafar. He changes back to Anakin in the final moments of the last movie when he sacrifices himself to save Luke.
Wookies are just one of the many species of aliens in the Star Wars galaxy. Tall, covered in hair, a Wookie is very strong and loyal. One example that most are familiar with is Chewbacca, Han's co-pilot.
I hope that helps. Don't tell someone you know something if you don't!