There are fundamentally two different ways in which something gets printed on paper or clothes:
Rasterized printing - there are many printing techniques that fall under this general category. While the quality and feel of the end result may differ between them, it boils down to printing pixels on either paper or clothes. The most common printer types that most consumers are familiar with such as inkjet and laser printers fall into this category. There are even industrial inkjet style printers that can print on clothes.
Even though the printing is fundamentally done with pixels, it still makes sense to use vector images as the input as they allow taking full advantage of the printer's resolution (see below).
Shape printing - again, there are many printing techniques that fall under this general category. While the quality and feel of the end result may differ between them, it boils down to cutting or extruding the shapes of the input vector image in some base material (silk screens, copper plates, other) that is then used to transfer the relevant color to the right place on the end product.
These processes typically require vector input to work at all - in most cases you can't cut/extrude the base material without the shape definitions used in vector graphics.
Printing has much higher resolution (typically 600 pixels/dots per inch, DPI) than what you see on a computer screen (typically 72 pixels/dots per inch, DPI). This means that while something might show just fine on your screen, it is going to look "grainy" or pixelated when actually printed.
See the Wikipedia articles on textile printing, flexography, screen printing, and digital printing.
There are tens of thousands of custom print shops around the globe. It would not be practical to list even a small fraction of these - for most printing needs we recommend checking your local yellow pages.
There are a number of larger websites that offer different types of printing on clothes. Some of these are:
These are listed for your reference and inclusion in this list does not constitute an endorsement by Vector Magic.
Signs come in all shapes and sizes and many of them are made using some form of vector image original.
While the specific production processes vary, one of them includes cutting the relevant colors from vinyl.
Computer-controlled sewing machines can quickly stitch custom designs onto different kinds of clothing. This process typically requires vector input to work at all - the machine needs to be steered to stitch out the relevant shapes and that can't be done using only pixel based information.
Graphic design is done in many ways with many tools, but the versatility and flexibility of vector images make them a very natural canvas to use when creating different forms of graphics such as logos or drawings.
A lot of the smooth animations and creative art you see in animated content online is done using vector images inside HTML5 or Adobe Flash animations.