The finest resolution you can print on paper right now depends both on the paper and the ink, which is one reason many printer makers insist you use their paper for best results.
An ink jet printer shoots a tiny droplet of water-based liquid ink (actually, very dilute with lots of water) onto the paper surface. When it hits, the water with its dyes included starts to soak into the surface in all directions, so its diameter grows. The chemical details of the ink and the paper surface are set up deliberately for two important things. One is that the paper surface must absorb the water quickly in as small an area as possible so that it does not spread out or, worse, stay liquid long enough to be smeared by physical contact. Attached to that is the requirement that the paper wetted by all this water should not expand locally and end up looking wrinkled. The other thing is that the dyes carried in the water should be preferentially attached to the first paper surface component they encounter so that they are no longer being carried along by the flow of water into the paper. This ensures the resulting dye dot is as small as possible. It also ensures the dyes are concentrated right on the surface to have maximum impact on the light it reflects back to your eye, thus yielding sharper colors and fine detail in the image.
For these purposes there are quite a range of papers available for ink jet printers. The simplest ones basically have a light treatment of starch on the surface, just like almost all office copy papers. That's why you can "get away with" using plain copier papers in an ink jet printer - it sort of meets the basic requirements of the printing process, but not well. The next step up is more sophisticated surface coatings with various pigment systems, although you would not see these coatings and probably don't even know they are there. These are much better than plain copier paper, but they look the same and cause confusion and skepticism among paper buyers - why buy more expensive "ink jet paper" when it looks just the same? Well, it actually is different and performance is noticeably better if you care to look at it. The next step up from there is a much higher-quality surface coating based on silica pigments, and these are the original "ink jet papers". The coating materials and process are expensive but produce great printed results. Again, this paper is not obvious because the coating is not glossy, and that's what most people think of when we say "coated paper".
Above that are a big range of multiply-coated papers for higher-quality results, often sold as photo printing papers. These come in both matte and glossy finishes, and they are optimized to accept lots of ink (with lots of water), to keep the ink dyes on the surface, and to minimize spread or "blotting" of the ink dot for sharp detail. Because optimization depends on the interaction of ink and paper surface, best results are obtained if you get both from the same manufacturer since each uses a different system.
Because the limiting factor on an ink jet printer is the process of liquid droplet spreading on the paper surface before its dyes are immobilized, the current limit appears to be about 1200 dpi per color. I have not seen any consumer-level printer that claims more, although there may be some at the professional level.
With the dry toner systems of laser printers the process and limits are different. The process involves creating an "image" composed of electrical charges on a drum surface, then attracting to that some fine pigment particles from a colored dust, then transferring that array of pigment particles onto the paper surface, and finally heating the particles to melt a fine layer of adhesive on the pigment particles so that, when they cool again, they are bound to each other and to the paper surface. This involves a lot of careful balancing of electrical performance of pigment, drum and paper. The pigment particles have to be small and uniform in size. The paper surface does require some modification with simple coatings to balance its electrical conductivity in the right range. The paper body itself must stand up to the "mistreatment" (as papermakers might call it) of being heated drastically but briefly on one side only, and still stay flat so it can continue to travel through the printer's paper path. To do full color, most such printers will do this process four times in sequence (one pass for each color), so the pigments are being laid down over top of a previous pigment and the paper is being hit hard four times. Moreover, each color's image has to be placed exactly right in relation to the other three for details to come out right.
So limits are imposed by toner particle size, laser beam and scanner resolution, balanced electrical characteristics of ink, drum and paper surface, stability of the paper body, and precision of the paper carrying system. The highest resolution I've seen for ONE color laser printers is about 4800 dpi - don't know if there are any professional systems higher. For 4-color laser printers, I doubt you'll see better than 2400 dpi per color.
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