Ah yes, the pen tool is your friend.
creating lines like this is rather simple.
Select the pen tool,
click once on your art board,
click somewhere else on your artboard (but this time hold and drag to create a curved line)
Let go of the mouse button, when you are satisfied with the curve of the line.
You should notice a "handle" that is attatched to the last point you just made.
(If you were to click again, a funky curved line would appear much to your disliking)
So, click back on your previous point to deactivate it wanting to make the next point interact further with the curve.
Make your next point close to the original point...click, drag, let go.
Click back on the point, to deactivate the curve again...
Click on original point to close your path.
Sums up to...
Click, click drag...let go, click on second point, click drag...let go, click back on third point, click original point.
This should have given you the top part of the image your wanting to recreate...you can draw new paths, and combine them (with the pathfinder tool)
This process can be used to make more complex things...and in combination with the pathfinder tool (as mentioned), you can have loads of fun.
Hope that made sense.