Why is a 1mW laser dangerous? In our Physics lab we have a 1 milliwatt (0.001W) helium neon laser. Despite the low power, we were cautioned not to even look at reflections of the beam as it could cause permanent eye damage - why is such a low power level hazardous? I have a 3W LED which I can look directly into which is more than 3,000x more powerful.
 A: As far as I know you have to worry about how much energy being deposited per surface area. And the area of the "hotspot" of the laser can be very small. So the deposited energy it enough to kill cells on your retina.
Actually there is a whole article on Wikipedia about it.
A: Because the light from the laser is collimated. This means that you have all this energy focused on a small spot. The larger power from a non collimated light source is spatially spread and the effective amount of energy that reaches your retina per time is much smaller.
A: The laser is most likely not dangerous at all when used normally, because it is very low power and within the visible portion of the spectrum.   The idea is that it is sufficiently low power that your blink reflex will protect your eye before any damage occurs.
The danger of a given laser source depends on its power, power density, and wavelength, as these determine what part of the eye will be damaged, how quickly, and whether you will blink.  The Wikipedia article on Laser Safety gives a fairly comprehensive summary of the matter.
As others have written, the main reason why lasers are dangerous in general is because they can concentrate a large amount of power into a small area.  The school most likely wants to instill a sense of respect for lasers and inspire good operating habits (both admirable goals) even though in this case they are probably over-stating the danger.
Source: I have to take a laser safety course every year for my job.  I work with lasers every day.
A: Dear Thomas, the diameter of the beams of these HeNe lasers is between 0.5 and 1 millimeter, so the power 1 mW is coming to $10^{-6}$ squared meters or so. The ratio of power and area is $10^{-3}/10^{-6} = 10^{+3}$ Watts per squared meter.
On the other hand, when a 3W LED is watched from the distance 0.1 meters, the power of 3 W is divided to $4\pi R^2 = 0.13$ squared meters, so the power per units area is 3/0.13 = 23 Watts per squared meter. In both cases, the eye will refocus the beam so the power per unit area of the retina will actually be much higher in both cases.
In this estimate, the laser has about 40 times higher power per unit area than the LED light bulb. Of course, it's small, so it will only burn one small point in your retina, but that's a bad enough problem. ;-)
A: A laser is coherent, collimate and intense
It may be a small amount of power but all that comes down a precise beam, in the same phase, at the same time. Your eye is a lens and focuses it into an even smaller spot, burning the retina.
Try staring at a 200w bulb for 1 minute, note the temporary black spots when you look away. That 200w is radiating in all directions, is incoherent, and wastes a huge amount as heat. The 0.1mw laser delivers more in a shorter time period to your retina than the bulb will
A: 1mW is not enough to do damage, its more of just an annoying distraction. Going to middle school then high school, 1997-1998 they ended up being banned because of people shining lasers into other peoples faces. Once you go beyond 5mW the blink reflex no longer offers protection.
I perminently damaged the CCD chip of a $200 digital camera with a 30mW green laser. I own a 1,100mW blue laser so powerfull, the dot on a wall is bright as the sun to a digital camera, and without eye protection, blue light is espescially dangerous at high levels because of photo chemical reactions in the eye, even just looking at the dot. 1,100mW will burn skin instantly, and even start fires, in the eye, tissue will get hot enough to boil the fluid likely as 1,100mW will boil coffee at point of beam contact. And I shined the beam on a drop of water on wood, and the drop sizzled away. 
Lasers can be dangerous, but the 1mW red diode lasers you can get for $2 are harmless.
A: normal "incoherent" light (white light from a bulb for example) is a world apart from coherent laser (Light Amplification by Stimulated Emission of Radiation) light
white light is made up of many frequencies (colors) of light, moving outward in all directions from a point source
laser light is made up of ONE color (read frequency) of light, moving in ONE direction, from a point source
to draw a simile,,,, 
white light is kinda like a crowd in motion,, many people, wearing different colored clothing, moving in different directions & all out of step with each other
laser light is kinda like an army, all wearing the same color clothing, all walking in the same direction & all walking in step with each other
now think of the "power" difference between a crowd & an army - a small army has a great deal more power than a large crowd - not because of numbers,, but because the army is focussed & directed, rather than scattered
similarly, a tiny amount of laser light has a great deal more "power" than normal light
A: If you have ever looked at a laser beam expanded and projected on a screen you may have noticed it has a speckled appearance.  A red HeNe beam spot after it passes through a divergent lens is not a uniform illumination.  Those spots are little regions with much higher irradiance.  If that beam goes directly into the eye, where the lens focuses the light, some retinal cells can be zapped.  If you get even low power laser beam directly into the eye with considerable repetition you can accumulate retinal damage.
Of course now this is compounded by the green lasers of considerably higher power that knuckleheads use to play as Starwars light sabers or worse to flash at aircraft.  These are only to point at the sky, where it reflects off the sodium layer and lets one point out stars etc
