Lines will be thicker where there's less light, tapering gradually towards more lighted areas, often attenuating to near-invisibility or vanishing entirely in the areas with the strongest light. (For a good example of this, look at Japanese manga and the wispy styles they often use.) A round object can gain greater depth if the line weight is heavier on the side away from the light and thinner on the side towards the light, giving a sense of three-dimensionality. A heavy object in motion can use line weight to convey the direction of motion and where the weight of the object redistributes because of gravity, squash, and stretch.
The simple answer to how to create this effect is "draw thicker and thinner lines." You don't need any special tools for this; just a drawing/painting utensil and your eyes. But there are still multiple ways you can vary line weight to create unique effects, or adapt to the tools at hand.
The easiest way to vary line weight is through pressure. Whether you're using a pencil, pen, marker, or tablet pen, controlling the pressure of the tool on the page will cause you to draw lighter or heavier lines. Learning varied and fine control can let you draw seamless lines that smoothly blend between heavier and lighter line weights without ever lifting the pencil.
You can also vary the angle of your pencil or pen; certain tablet pens have this setting programmed in as well, depending on the software you're working in. Varying the angle causes you to draw more with the smaller, finer point or more with the side of the pen or pencil; the point creates lighter, sharper lines, while the side will create thicker, softer lines, sometimes smudgy or messy. This works with blunting the point of a pencil, as well. A freshly-sharpened pencil will draw thinner, scratchier lines than one that's been blunted and softened to a duller point.
If you have trouble varying using pressure and tilt, you can retrace. Draw in a uniform, light line weight at first, then retrace to thicken the lines in the right areas, going over it as many times as you need and carefully blending the lines together to make them taper smoothly.
You can also vary your tools. A 2B pencil makes a different type of line than a 2H pencil; a ball-point pen makes a different type of line from a felt-tip pen, which is wholly different from a market, a liquid-gel pen, or a calligraphy pen, which by its very nature is intended to create elegant lines of varied weights. Play with harder and softer leads, pens with harder and softer tips or lighter or heavier ink flow. Real quill pens are especially fun and fascinating to work with.
If you're painting in programs like Photoshop, using a calligraphic brush can help you automatically create line weight. Calligraphic brushes use a narrow ovular brush shape tilted at an angle to mimic the nib of a calligraphy pen, but you do have to be careful not to fall back on the brush defaults and paint in line weights that don't match the directions, highlights, and shadows in your scenes.
In vector programs, creating line weight can be a little more difficult. While some come with calligraphic brushes just like Photoshop, others will only let you vary the line style, but not the line weight - so that even if you change the appearance of the line overall, it stays the same thickness all the way through. There are ways around this; just like with a pencil or pen, you can use the thinnest line possible for your initial outline, then go back and retrace areas to subtly thicken them as many times as you need. What you can also do, though, is treat the lines like their own shape, with their own outline, and draw them on using vector curves, vertex points, and handles that allow you to create filled areas of varying thickness for your outlines.
Whatever you do, make sure the variations in line weight are used to consistently flatter and enhance your art, to give it a more finished, polished look.
