To fall, you need to have your tiles designated as "solid" and "not solid". When your character is over a "not solid" tile, you move him in the negative y direction while moving him in the x direction according to key presses (assuming you're using key presses to control) as usual. Note that this is the mario-specific way to do things, some games don't allow you to change the direction of your fall, they just calculated the angle of fall when you hit the edge and drop you that way.
For jumping it's more or less the same idea as falling, except for the fact that you go in the positive y direction until you reach a certain height, then you go in the negative y direction until you are back on a "solid" tile. Make sure that the positive amount of y you jump is relative to your starting y, otherwise you will be unable to jump over a certain height regardless of where you are on screen if you do it relative to a set screen bottom, or if you do it relative to the "solid" tile you are currently over your character will get a strange tendency to float up stairs.

Hope this helps, g'luck
-fel