How to I get clear images with my nikon d5000?

So I'm in school for digital photography. Last week my teacher was talking about getting clear enough pictures that you could zoom in and see stuff as eyelashes but said he's not getting into that this class. When I zoom into my pictures close enough to the eyelash it just gets blurry. How do I make them that clear?

There are many factors that go into the creation of a tack-sharp picture.

First off, I suspect you need a dedicated macro lens to get as close as it sounds like you want to here. Macro lenses are for very close-up work.

Secondly, you need to really nail the focus. Being even the tiniest bit off will ruin the shot if you're going for perfect sharpness. For this reason, most serious macro photographers use a tripod 99% of the time - it holds the camera in one place so tiny movements don't throw off the critical focus.

Finally, you need a shutter speed that is fast enough that motion blur isn't a problem. That will depend from subject to subject and from lens to lens, but for people, try to stay above 1/100 of second at LEAST if you want them to have no motion blur. Faster is better.

There are a lot of other "little" things that can help you get that extra oomph of sharpness in a photo: mirror lockup, using the sweet spot on the lens's aperture, using only the highest quality optics.etc., but if you don't have the basics, then none of these "little" things will help anyway.

Probably because you are not holding the camera firmly, it needs to be rock steady whilst the shutter is open, when shooting in low light the longer shutter time will make this worse.

Use a tripod will eliminate any camera shake, shooting in bright light will reduce shutter time.

Flash can help too, the flash duration will be faster than the shutter time, if most of the light from the image comes from the flash then your effective shutter speed is the very fast flash duration, usually faster than 1/1000th of a second or faster.

It's simply the minimum focusing distance of the camera lens. They can only focus so close, closer than that and the pictures are blurry. You can buy a macro lens or macro diopter filters to place on the front of your current lens to be able to focus as close as an eyelash. At this point, it's really your camera that's the problem.