Two incorrect understandings I've seen are:

• After 1 half-life, exactly half of all particles will have decayed
• After 2 half-lives, none of the original particles remain

To try to make this clear, I've simulated particles decaying. The plot below shows a simulation for 150 particles decaying with two different half-lives: 15 years and 40 years

As you'd expect, 50% are gone after one half-life, 75% are gone after two half-lives, etc. However, notice that it isn't a perfectly smooth or thin line. There is a spread in # of particles remaining after one half-life. Some simulations had more than 50% and some had less. 

A related concept is how radioactive the particles are. You'll often hear of elements with extremely long half-lives. Uranium 238's half-life is roughly 4.5 billion years for example. Does that mean that it would make an area uninhabitable for 4.5 billion years?

Below is the average radioactivity for the simulations from above. The y-axis has arbitrary units (# of particles in simulation that were emitted), but the scale is the same so that ratio the two curves is relevant:

Notice that the curve with the longer half-life is much less radioactive at the beginning. It is that way by definition...the more particles decay, the more radioactive it will be and the shorter the half-life is. Thus...if you're thinking in terms of danger, elements with half-lives somewhere near a human time scale are the worst for habitability of areas. If the half-life s 2 hours, it's super-radioactive right away, but you can just evacuate the area for a few days and wait for it to decay. If it's 4.5 billion years, the radioactivity will be low. If it's 10 years, then the area becomes uninhabitable.

Cesium-137 is an example of a problematic one...its half-life is ~30 years, and it is a primary concern with the Chernobyl region.

Anyway...really simple overview and there are multiple types of radiation and we are more susceptible to some elements than others, but it hopefully makes the concept of a half-life a bit clearer.