It's pretty involved, but a general rundown would go something like this:
SPOILERS
After reports of animal attacks in the woods around the city of Raccoon, a special squad is sent to investigate. What is eventually discovered is that the company Umbrella Incorporated has been secretly developing a virus for use as a weapon. This virus reanimates the dead and can be used to make hideous creatures referred to as "Bio-Organic Weapons" or BOWs. The captain of this special squad was actually put in place by Umbrella and lured them to the remote Spencer Mansion as part of a test to see the BOWs in action, but in the end, Wesker - the captain of S.T.A.R.S. - is defeated and the survivors (Chris Redfield and Jill Valentine among them) high tail it out of there.
Turns out this whole thing started back in the sixties with a man named Spencer, discovering a plant in Africa and he got the wonderful idea to create a new world where he could reign as a god. To hide his research he created Umbrella with the aid of cohorts and his nefarious plot developed.
After the 'mansion incident' the troubles didn't stop there. Because of a rogue scientist on Umbrella's payroll, an outbreak occurs in Raccoon City and before long, the whole place is infected and falls into chaos with zombies and strange creatures running around. Because of the scandal, the government decides to clean up the mess Michael Bay style and a warhead is dropped, blasting the city into oblivion. The survivors here include Leon Kennedy - a cop; and Claire Redfield - who was in town, looking for her brother; and Jill Valentine who stayed in town while Chris ran off to god knows where, chasing Umbrella.
Claire later joins the fight against Umbrella and goes overseas to take on their European headquarters and presumably find her brother, but is captured and taken to a prison colony. The prison colony is attacked and all sorts of Umbrella brand nasties are released. We also learn that Wesker is still alive. Not only that, but he's got funky powers now too. He can move fast and his eyes glow in the dark. All because of a funky 'new' virus. After Claire manages to mostly escape, and after sending a message to Leon about her whereabouts, Chris arrives looking for her and not only do they reunite, but Chris gets to come face to face with Wesker again. Wesker escapes, but so do our heroes.
Fast forward some years later and Leon Kennedy, one of the survivors of the Raccoon City incident is sent overseas to rescue the president's daughter. It's worth mentioning that Umbrella didn't do so well after it was uncovered that they were responsible for the outbreak in Raccoon City and are at this point dissolved, simply disappearing from the story.
Turns out the president's daughter has been kidnapped by a group with their own delusions of grandeur and even worse - are all infected with a strange parasite that lets one man control them all, while granting strange attributes to those infected. These are the Plagas. Leon now works for the US government and on top of his assignment to rescue the daughter, takes them all on and saves the day.
Then we come to RE5, where Chris is sent to Africa to stop a weapons deal and stumbles onto a plot that goes all the way back to the very beginning... to a man named Spencer. I'll leave out the rest so nothing is ruined for you.
That about sums it up - leaving out many of the sub plots of course.
