A quick search at ask.com gave lots of answers. Good old CCD-based (Charge Coupled Device) digital cameras can electronically capture images with no need for a shutter. In that case the shutter is called an electronic shutter because the beginning and ending of the light-capture moment is controlled electronically by software. However, due to the limitations of semiconductor electronics, the fastest they can go is 1/500th of a second. Think: you have to move packets of electronic charge from up to 10 million pixels all at once or risk distortion with a moving subject. In addition the mechanism might need to capture those 10 million pixels three separate times with each of red, green and blue filters. Often photographers need faster shutter speeds than 1/500th second to stop racing cars, raindrops, and other fast-moving objects. So while some low-end digital cameras, such as those found in cell phones, indeed have no mechanical shutter, high-end models have both electronic and electro-mechanical shutters so they can get down to speeds as low as 1/10,000th of a second.
Today, however, most consumer digital cameras employ the cheaper CMOS (Complementary Metal Oxide Semiconductor) technology in their imaging chips, not CCD technology. With CMOS, the fastest you can switch the pixels on and off is about 1/250th of a second. This is because these chips rely on a number of transistors for each pixel and each of these transistors must switch on and off. All of those switchings take time and it adds up when you have millions of them. So these cameras also need mechanical shutters to get to speeds of 1/1000 of a second, which is typical of ordinary consumer cameras.
Ironically, unlike film cameras the shutter in a digital camera is actually *always open*. Otherwise you would not be able to see the image on the back screen to frame the shot. At the instant of exposure, the shutter is closed, then opened for a fraction of a second to expose the image on the chip.
Here are some references:
http://www.dalsa.com/shared/content/Photonics_Spectra_CCDvsCMOS_Litwiller.pdf (page 2)