Or what are voltage and current exactly?
I have seen this explained with water, but that doesn't make any sense. Water doesn't have any volts!
Let's start with a simple case. You have a battery, and you have a light-bulb that you connect to it.
That is like a bottle of vodka and a drunkard. The voltage stored in the battery is like the strength of the booze in the bottle -- it's given by the chemistry of the particular kind of battery. The current, on the other hand, is given by how fast the drunkard drinks it. He can sip it slowly, and thus get only a little bit druk, while the bottle lasts long, or he can guzzle it straight from the bottle, which is the equivalent of making a short. Of course, the more he drinks, the hotter he gets. To get to the same level of inebriation with a lower voltage, you need to drink more of it, so if you want to do it in the same time, you need higher current. The slowness of drinking is the resistance.
The bottle has its own resistance too. The bottleneck is the bottle neck. You can only pour so fast from it. However, if you connect two or more bottles in parallel, you can get much more current from them. (Connecting bottles in series gives you higher voltage, which is where this metaphor breaks, but bear with me.)
Of course, if you have several drinkers in parallel, you will need higher current. (Again, the metaphor breaks when you connect them them in series, but let's not dwell on that.)
Now, what happens when the drunkard tries to drink faster than what the bottle can manage? In fact, a bottle is more like a capacitor than like a battery. It holds a certain amount of electricity at a certain voltage, and lets you draw all of it from it. (But not quite, to make it exactly like a capacitor, you have to top it up with water after every pouring, so that voltage drops.) A battery is more like a distillery. It produces as much vodka as needed (at a certain voltage), but when you try and demand more than it can produce, it will start topping it up with water, and thus you get a voltage drop. If you do it too much, the distillery will break.
Now, if you connect two distilleries in parallel, they will double the voltage of the resulting booze. If you connect them in parallel, you can draw more current.
That's pretty much it. The important thing to remember is that the voltage is determined by the distillery, but the current is determined by the drinkers (and limited by the distillery by voltage drops).