Since voltage determines an electron's energy, why is it current rather than voltage that is harmful? Electrons have more energy with higher voltage so it seems that voltage would be what actually determines how harmful it is, because it would have more energy to lose.
 A: The phrase "it's the current that kills, not the voltage" is vastly oversimplified and ambiguous.
James' answer details why current flowing through your body is a bad thing, so I'm going to focus on why I dislike this phrase.
For starters, to get a current to flow through your body in the first place, a sufficiently high voltage is necessary. After all, humans aren't incredibly conductive, so the electrons in your body aren't going to start moving without a reasonable potential. That's why you don't have to put on a bunch of protective gear every time you change your car battery-- even though it can easily output hundreds of amps, its paltry 12 volts can't push a dangerous current through your body (Also, as a word to the wise: low voltage, high current supplies can still cause injury. My electronics teacher was always fond of telling a story about how his friend was working on such a supply while wearing a ring, and at one point his wrench on one terminal touched the ring which was contacting the other terminal, and the resulting high current through the ring heated it up and caused him to lose the finger). So, my correction #1 to the phrase would be "but a voltage is required to induce a current".
But even this isn't the whole story, as different voltage sources behave differently under load. For instance, static electricity isn't usually dangerous because even though the voltage is high enough to push a significant current through your body, it very rapidly decreases as it discharges, and the time span is short enough to avoid any real damage. But if you go out and grab two different phases of a 480 VAC mains power source (note: don't do this), the line to line voltage will pretty much be unchanged by your addition to the circuit, and it will continue to push the same current through you indefinitely. So my addition #2 would be "Current alone does not guarantee lethality; the duration of this current and total energy delivered to your tissue also play a crucial role. Since different power sources behave differently under load, the danger of a given setup must be analyzed on a case by case basis."
As you might be able to tell, writing pithy rules of thumb isn't my strong suit :)
EDIT As freecharly pointed out, current in the body is primarily carried by ions, not electrons, and the low conductance has to do with how many of them there are. But the main point of the paragraph still holds: you can't induce a dangerous current flow through the body without a high enough voltage.
A: Voltage is analogous to force.  Just because a force is acting on an object, that does not necessarily mean that the object will move.
Current is motion:  It is the motion of charges, and in an ionic solution (e.g., in your body tissues), the charge carriers are ions. Current causes ions to move, which means, it causes chemical reactions to occur.
All of your life processes are chemical reactions, and the flowing current messes with that.  In particular, the function of your nervous system depends on the movement of ions.  Flowing current through your body tissues messes with that a lot.  If the nerve signals that are disrupted happen to be part of your body's cardiac pacemaker system, it could put that system into an abnormal state in which your heart does not beat.
That usually is bad. 
A: There are two main ways electricity harms the body. First, the human nervous system uses electrical signals to communicate. An external source of electricity can hijack this messaging system. If cardiac signals are disrupted, this can be fatal.
Electricity also can harm through raw power; as electricity flows through the body, energy is dissipated by the current and absorbed by the body. This can cause serious burns.
The first type of damage can occur with very low voltages, but is highly dependent on the location; being tazed in the foot is lot less dangerous than being tazed in the chest. The second depends on the power dissipated, and so both current and voltage are important, but current is largely determined by voltage. 
You can also have a combination; an electrical current can cause muscles to spasm, making it impossible to let go of a wire, causing burns.
