The stage builder requires the Game Pad. You can design stages using the touch screen and stylus, placing various objects and obstacles too.
You can choose from a variety of backgrounds, and any song you've unlocked. These are just aesthetic differences, of course.
More importantly, you can choose the size of your stage. You'll create the ground yourself, but the box you put it in can be small, medium, or large, which greatly affects where the edge of the screen is.
Once you start designing, zoom out and take note of what you see. The "ghost" island in the middle of the screen is a general guide for where to put your stage, though there's no rule that says it has to go exactly there. The green box shows the boundaries of the camera. You'll never see past the green lines when playing. The dark outer box is the blast zone. You can't place anything here, and if you touch here, you'll be KO'd.
Along the bottom, you'll see your tools. Note the "Weight" meter. This meter will fill as you add things, and you won't be able to add anything else once it's full.
The line icon will allow you to draw ground. Either draw it freehand like with a pen, or tap points on the screen to draw straight lines. Draw a shape to create thick ground, or a line to create a thin platform that you can pass through. (To "complete" a line, touch one of the icons on the bottom of the screen.) By touching the line icon when you're not drawing, you can choose what your ground will look like. This is for appearances only.
If you see a little blue "+" on the edge of a platform, you can decide whether that edge will be grabbable or not. Touch the + to change this.
Press X to make a grid appear. This will help you judge exact distances. While the grid is shown, you'll only be able to draw straight horizontal and vertical lines. This can come in handy, or get in the way, depending on what you're trying to do.
The eraser icon will erase whatever it touches. Unfortunately, you can't simply erase part of an object or platform. You have to erase the whole thing.
The left icon holds various objects with special functions.
Springs will cause characters to bounce. Large springs are easier to hit and will send you farther than small ones.
Similarly, you can launch yourself out of cannons, and the large ones will send you farther than small ones. Once you place a cannon, you can use the red arrow to aim it.
The moving platforms also come in small and large sizes. Place the platform and then move the yellow rectangle to decide where it will move to.
Magma is a bit more complicated. You can't place magma directly, but you can turn part of all of an object you've drawn into magma by painting over it. Magma will damage anyone who touches it, but it's not a Danger Zone, so it won't KO anyone.
You can give your stage a name and then play on it in Smash mode. Get creative. Maybe make a stage shaped like a character or sprite. Create a stage that consists entirely of moving platforms. Make one shaped like a box, with magma on the top, so character will only spawn inside the box. This is terrible for regular matches, but great for Stamina matches, as you can create a stage where you definitely won't be KO'd with a ring out. Use your imagination!