I'll attempt to illustrate here with an example.
If you take one of the little 12V garage door opener batteries and short out (directly connect) the two terminals with a piece of wire or something else. You'll get a light current flow through the wire or metal. It may get a little warm. This battery is only capable of supplying a small amount of current.
If you take a 12V car battery and short out the two terminals (don't do it, it's not fun), you will be met with a huge current arc that will likely leave a burn mark on whatever was used to short it. This is because the car battery is capable of discharging a large amount of current in a very short period of time.
One time when working under the hood of a car I shorted out a wrench from the positive terminal of the battery to my watch, and my watch was touching the frame. In the brief instant the current was flowing through my watch it became glowing red hot and the watch left a burn mark (still scarred to this day) on my wrist. A 12V garage door opener battery would not have done this.
However, this is not the entire story. As suggested already by others, Ohm's Law describes the relationship between Voltage(V) and Current (I) with V=IR.
There is a great All About Circuits article that actually addresses your question here. They make a good point that if Voltage wasn't dangerous, nobody would ever make signs that said DANGER -- HIGH VOLTAGE!
They provide a great summary themselves at the end, which I'll provide here:
Harm to the body is a function of the amount of shock current. Higher voltage allows for the production of higher, more dangerous currents. Resistance opposes current, making high resistance a good protective measure against shock.
Any voltage above 30 is generally considered to be capable of delivering dangerous shock currents.
Metal jewelry is definitely bad to wear when working around electric circuits. Rings, watchbands, necklaces, bracelets, and other such adornments provide excellent electrical contact with your body, and can conduct current themselves enough to produce skin burns, even with low voltages.
Low voltages can still be dangerous even if they're too low to directly cause shock injury. They may be enough to startle the victim, causing them to jerk back and contact something more dangerous in the near vicinity.
When necessary to work on a "live" circuit, it is best to perform the work with one hand so as to prevent a deadly hand-to-hand (through the chest) shock current path.