A Path is a vector object that can be used to make selections, or as a target for a stroke or fill, and possibly other purposes. A default path called the "work path" will be created as soon as you start using the Pen Tool (P). All uses of the Pen Tool will add to this path until it is deselected in the Paths palette. If you deselect it and start using the Pen tool again, the work path will be overwritten. If you want to save that path, just drag the work path in the palette to the New Path icon, and it will be renamed and saved. At any time you can click the New Path button to create a new path, and whichever path you have highlighted in the palette will be the one you're working with. Okay, on to use of the Pen tool.
The Pen creates Bezier (pronounced BEZ-ee-ay; it's a French name) curves or splines, sometimes known as B-splines. These splines are made up of anchor points, each of which has up to two control handles, and the anchors are connected by the curve itself. The curve's shape is controlled by the angle and length of the control handles. In a Bezier spline, the line will always pass through the center of the anchor point, and the handles will always be tangent to the curve as it passes through the point. But that's quite enough theory. How do you actually draw something with the thing?
It takes some practice to get used to drawing with splines. With a Bezier, the trick is to put your anchor points in places where the line needs to change direction, and to use as few as possible. In the following images, I'll trace a photograph, so you can see where I place my anchors and what I do to get the shapes that I want.
In this image, all I need is an oval, and I want all of my points to be symmetrical, so I just click-drag out my lines and adjust them as I go so that the curve follows the mouth of the cement mixer:
I only selected one anchor point here, so you can only see the handles influencing that portion of the curve. The others look pretty much the same.
In the next image, my shape is a bit more complex. Notice that the anchors are only in the corners, and that I've broken the tangents so that the curves have a different trajectory leaving the anchor than they do entering it. Also notice that there are only handles on two of the points. The others are "cusp points," which means that the anchor does not influence the direction of the curve. Also notice that one point has only one handle. That was my starting point, and I did a simple click to start, then click-dragged to finish it, so it only got one handle. In order to break a tangent, click-drag to make the point and pull out its handles, then hold down the alt key. The handle furthest from the cursor will stay put, but the one nearer the cursor can be dragged around independently.
The nice thing about these vector lines is that you don't have to get them right from the first click. You can use the Direct Selection Tool (a) to select a point and move it. Once selected, that point's handles will appear, and you can manipulate them also. Again, to break the tangents, just hold down alt while dragging a handle. If you need to turn a smooth point into a cusp or vice-versa, you can use the Convert Point tool, which is in the Pen tool's flyout.
The best way, in my opinion, to learn how to use the Pen tool is to trace some images with it. Use it to make paths that you use to cut an object out of its background, and you'll quickly learn where to place your points and how to manipulate them for best effect. Start with something simple, like a piece of fruit, then do a car, and then something more complex like that cement mixer. If you want to really go nuts, get After Effects and start doing it with video; that's called rotoscoping, and it's one of the most essential tasks necessary when creating visual effects for film and TV (which is what I do for a living). And, of course, the Pen is the primary tool in vector illustration programs like Inkscape and Illustrator.
The photograph of the cement mixer is my own. The full image is available here: http://www.bryanray.name/photos/CementMixer.jpg