Sorry, I didn't read the above 2 posts, so I might repeat something, but I'll try to keep this short and intuitive.
As electrons travel from one plate of the cap to the other (through the battery, not the dialectric), a voltage builds up across the plates. That voltage acts against the voltage created by the battery. So, when the cap has no charge, voltage is greatest and therefore current is greatest (V=IR). When there is a higher current, the voltage across the cap rises faster (more electrons travel from one plate to another). Then as the voltage across the cap rises, current decreases and therefore the rate at which it charges decreases. Eventually the voltage across the cap is so close to the battery voltage that it is basically 0 and there is no (well, very little) current. Hence you get exponential curves.
If you do the math, you get T= RC where T is the timing constant, R is resistance in ohms (more ohms = less current = more time for constant C), and C is capacitance in farads.
This applet might help too:
http://www.phy.ntnu.edu.tw/java/rc/rc.html