Resolution: how does it work?
When dealing with digital files, you hear the word "resolution" come up a lot. It can be confusing, so for a full breakdown, check out this article.* But for the TL:DR version, here's the Clift's Notes version:
Resolution is a number that describes either the number of dots in a printed 1" x 1" area, or the number of pixels in a 1" x 1" area on a screen. Higher resolution files provide a sharper image and a larger file size. Lower resolution files allow for a smaller file with poorer image quality.
A good resolution range for printed images is 240-300dpi (dots per inch). Because computer screens render at 72ppi** (pixels per inch)—that's the resolution your eye can discern—images for digital applications can be lower resolution and therefore smaller, which allows web pages to load faster.
Resolution only applies to RASTER images—that is, images made up of pixels, like photos. Raster images are subject to pixellation and fuzziness when enlarged, because you're enlarging the pixels themselves (though raster images can reduce with no loss).
VECTOR images—many logos and illustrations, created in a vector drawing app like Adobe Illustrator—are composed of smooth, solid shapes, so they can enlarge and reduce with no loss of clarity or crispness. Resolution does not apply to vector art.
Many digital images are taken at a low resolution (72 dpi) but a large physical size. Reducing the physical size increases the resolution to achieve a smaller but sharper image. The larger the MP (megapixels) on a camera, the larger high-quality image you can produce.
Finally, in terms of screen printing, we ideally want to work with vector art whenever possible, so keep that in mind when we ask you for logo art or other assets—they provide a clean line that can be enlarged or reduced without loss of clarity, so your print design is as sharp as possible.
* I thought this would be a summary, but I think this is just as good as the article I linked to. Possibly better! ** Mac Retina screens actually display at 144ppi—double the standard computer display—but it's still much lower resolution than the printed page.