Drawing in perspective is much easier than you might imagine and it's a lot of fun. We'll start with simple onepoint perspective, see what it looks like, and practice constructing simple shapes.

The Concept of Perspective Drawing
The first thing you need to know is that in perspective drawing every set of parallel lines has its own vanishing point. That will make more sense in a moment. Remember from math class that parallel means running sidebyside, the same distance apart. This means that the sides of a road or the sides of a door can both be thought of as pairs of parallel lines.
Let's look at this picture. It shows a onepoint perspective view. All of the lines that are parallel to the horizon (at right angles to the direction of our gaze) such as the railway sleepers and fence posts—go straight across or straight up and down. If they were longer, they'd keep going straight across, or straight up and down. These lines will always stay the same distance apart and never meet each other.
In contrast, the lines moving away from us appear to get closer together as they get more distant. These lines meet at a vanishing point in the middle distance of the picture.
To draw using onepoint perspective, we arrange our view of the subject so that one set of visible lines has a vanishing point right in front of us. At the same time, the set at rightangles goes out to infinity on each side. So if it's a road, it goes straight away from us, or if it is a house, one wall goes straight across in front of us, not sloping.
In reality, of course, there are always objects which won't be lined up perfectly. For now, let's keep things simple.

OnePoint Perspective in Real Life
To make sense of what we will be drawing, let's first take a look at a box from a onepoint perspective in real life. Then we can see how it works.
Here's a photograph of a box on a table. Again, it shows us how one set of lines remain parallel and the other set vanishes to a point.
Note that the line across the back is not the horizon line. It's the edge of the table and is lower than my eye level, and so, lower than the horizon.
If we continue the lines made by the edges of the box, they meet at a point above the table and this is at eye level. Were we able to see into the distance, this vanishing point would be on the horizon. At the same time, notice how the front edges of the box are quite parallel.

Draw a Box in OnePoint Perspective
Let's draw a simple box using onepoint perspective.
 Draw a horizon line about onethird down your page.
 Use a small dot or line to mark a spot roughly in the middle of the line. That's your vanishing point.
Note: Don't make your vanishing point as big as this example. You want it to be small so that all your lines finish in exactly the same spot.

Starting the Box
 Draw a square or rectangle well below and to one side of your vanishing point.
 Make sure your vertical lines are perpendicular (at right angles) to your horizon line, and your horizontal lines are parallel.
No funny angles or wobbly lines! For a successful perspective drawing, you need straight lines and corners that meet exactly. If needed, use a ruler to ensure your lines are perfectly straight.
Continue to 5 of 10 below. 
Drawing the Orthogonals
 Draw a line from each corner of your square or rectangle to the vanishing point.
 Make sure they are straight and finish exactly at the vanishing point.
In perspective drawing, we call these lines orthogonal lines or orthogonals. These words derive (somewhat) from their meaning in mathematics because they are at right angles to the horizontal plane.

Continuing to Construct the Box
Now comes the tricky bit.
 Draw a horizontal line, starting a little way along the bottom left vanishing line, across until it joins the bottom right vanishing line. This is the bottom edge of the back of your box. Make sure it is straight and parallel to the horizon and front edge.
 Draw two vertical lines, straight up, from where that back line meets the two vanishing lines, up to the two top vanishing lines.
 Add the horizontal line that joins the two verticals you just drew.
The two biggest problems at this stage of the drawing are lines at angles—they must be straight—and lines that don't quite meet. If you stop short or go past the vanishing line ever so slightly, with one of the lines, you'll have trouble getting your last line straight.
If your box is close to the horizon or vanishing point, you might find that the angles are very obtuse (wide) and hard to get right.

Clean It up and Finish the Box
 Now that you have finished constructing the box, you can erase the vanishing lines.
 You can keep all the lines inside the box if you want it to be seethrough, like a fish tank.
 Or, you can carefully erase the back corner—the bottom left, back, and lower back lines, as you see in the example.

Multiple Shapes in OnePoint Perspective
Let's take a look at a few more examples of onepoint perspective drawings. Why not have a go at drawing some of these yourself? Several objects on a single page can look very cool.
 Try drawing more squares and rectangles in different places.
 Place one above the horizon line and one right in the middle below the vanishing point so you can see the effects.
Continue to 9 of 10 below. 
Draw the Vanishing Lines
 Draw each of the vanishing lines for your boxes.
 Remember that all vanishing lines need to go to the vanishing point, even those above the horizon line.
As long as your ruler is lined up correctly, you can stop drawing just short of the vanishing point. This will make it is easy to see and the vanishing point won't get lost in a tangle of lines.

Complete the Single Point Perspective Lesson
 Finish off your single point boxes by completing the boxes.
 Erase your excess vanishing lines.
To get more practice with perspective drawing, try constructing some simple boxes and making them into complete drawings. You could draw a fish tank, an open box, and a solid box. Experiment with putting your horizon line at different heights.