here's the wiki on risk. i've been playing for quite awhile and this seems pretty straight forward. the hardest part is figuring out the numerical value of each piece if you have the version with the little figures or the little blocks. there is a version that just has little roman numerals.
basically you choose a color and take turns with the other player(s) and place your pieces in the various countries until all territories are occupied.
then you choose an opponent's country to invade and take turns rolling 2 standard die.
each time you win a roll you take away one of your opponent's army's. you keep going until one or the other's armies in a territory have been completely diminished or you decide to stop attacking that particular territory.
you may only attack adjacent territories, i.e, those that share a border or are connected by a dotted line. so you can't attack italy from south america.
the game continues like this until one person controls the entire board.
a couple of hints i've found that work great:
the key territories to control are greenland, argentina, australia, and madagascar. they make great strongholds and can attack a variety of positions.
consult the website for some more detailed directions. i think there's a few minor details i might have left out but this is the gist of it. good luck and have fun!

The direct link to the Hasbro rules page for games starting with "R" is http://www.hasbro.com/default.cfm?page=cs_instructions&letter=R

Scan the page to find the version of Risk that you own, then click on the link to download a PDF of the rules.

Eric Martin
Editor, http://www.BoardgameNews.com