The Wikipedia article on Laser Safety completely covers how lasers damage the eye, what types of lasers there are (in terms of damaging potential), and other related topics. So I'll give the brief summary of relevant info here.
There are 4 main classifications of lasers (aptly named class 1 through class 4).
Class 1 lasers are completely safe. This is usually because the laser beam itself is hidden, but for class 1 lasers with open beams, you can stare into them as long as you want without damaging your eye (unless you do something stupid like focus the beam with a telescope).
Class 2 lasers are safe because the time it would take to damage your eye if looking directly into it is greater than the time it takes for your brain to say "Ah, bright light! Blink, you dummy!" and then carry out the blink, thus protecting your eye. Again, don't be stupid and look at the beam through a magnifying glass or something.
Class 3 lasers should be handled carefully. They can damage the eye if you directly view the beam (although the low-power end of this classification carries a low risk of eye damage). It is perfectly safe to view the diffuse reflection of a class 3 laser (that is a reflection off a non-shiny surface. It is also perfectly safe to look at the beam indirectly (such as using mist or smoke to illuminate the beam path). Protective goggles are a recommendation when there's a chance of directly viewing the beam (that is, putting your eye right along the path of it).
Class 4 lasers are the most dangerous. Anything too powerful to be a class 3 is listed as a class 4. You can damage your eyes by looking at a class 4 laser beam in any way (directly, indirectly, even looking at the laser dot on a surface can damage your eyes). Protective goggles are always required. Class 4 lasers can even burn your skin or clothes and ignite combustible materials. There is little chance of most people encountering a class 4 laser, but if you do, make sure you are protected, don't walk in front of the beam, and most importantly, don't be stupid.
The main factor in determining the class of a laser is its output power. Class 2 lasers don't go higher than $1~\rm mW$. Class 3 lasers are separated into 2 subcategories; 3R for lasers below $5~\rm mW$, and 3B for lasers up to a continuous beam power of $500~\rm mW$. Above $500~\rm mW$ is class 4 lasers. I've worked with a few class 4 lasers myself, the lowest power one was $700~\rm mW$ and the beam was warm to the touch even then. So above $500~\rm mW$, no-touchy.