There are a lot of solutions to the Rubik's Cube. Many of them are complicated. Maybe this one will be to you. But for me, this is the solution that is easiest to understand and to remember. It's definitely not optimal either in terms of time to completion nor in terms of moves to completion. It consists of seven stages and six algorithms.
An algorithm might look something like: D L D' L' D' F' D F. Each letter
means to turn the corresponding face one rotation. Clockwise if there is no
tic mark, counterclockwise if there is. Each face has a middle piece which
defines the color for that face, four edge pieces which form a cross with the
middle piece, and four corner pieces which form the diagonals across that
face. The faces are:
Second layer edges
Now you need to put the second layer edges into their correct positions.
When you are done with this stage, your cube should look like this:

Figure out which of these looks like your Upper face and use this
algorithm to change it to next image on the right. Repeat until
you have a cross as in the final image on the right.


[The Front face for these images is hanging off the left side.] 
Upper layer edges position
Now you need to rotate the edges until they are in their proper positions (i.e.
aligned with the middle pieces in the second row). What you do is spin
your upper face until you have one edge correctly lined up (put that on your
F face), and the other three need to rotate counterclockwise. You may need
to do this either zero times if you're lucky, one time, or multiple times if
the edges need more than one rotation.
R U R' U R U U R' 
Upper layer corners position
Now that you have your Upper cross in place, all that's left is the Upper corners. First you rotate them until they are in their correct positions. They will
probably be twisted (i.e. misoriented) but that's the next stage. Hopefully,
you'll start with at least one corner in its correct location. Rotate your
entire cube to make that corner be the ULB corner. Then use the following
algorithm to rotate the other three corners clockwise. You might have to
do this multiple times.
R' U L U' R U L' U' 
Upper layer corners orient
Almost done. Now you just need to twist any corners that aren't already
correctly orientied. Start by rotating your entire cube until you have
a misoriented corner at LFU. Do R' D' R D multiple times until
that corner is correctly oriented. Then do U (i.e. just spin your upper face) until another misoriented corner is at LFU. Repeat this until you have
all corners correctly oriented. If you have multiple misoriented corners, your
entire cube might looked all messed up until you finally correctly orient all
of them. Don't worry about it; just keep going. Once you get all the
corners correctly oriented, you might need to just spin U or D once or twice
to completely solve the cube. The complete algorithm for this stage (in
computer programming pseudocode) is as follows:
while misoriented corners exist { U until bad corner at RFU do ( R ' D' R D ) until original RFU correctly oriented } U and/or D to completely solve 
Congrats! You've solved it. Here's a fun page that shows how to make various patterns from a solved cube: pattern page.
Here's a little cheat sheet with all of this same info that my friend Ben McClelland wrote: , or as a PDF.