Tiny dots or pixels are measured by their density in an image. A megapixel is 1 million pixels. The more you have, the more digital information is contained within the photo and the better it can look. That said, you don't want to choose a digital camera with less than 3 megapixels. In fact, you might want to go for 4 megapixels or higher. At that quality, pictures look great online and can be printed bigger than an 8X10 size without any noticeable loss in detail.
The good news is that digital cameras have come way down in price, and 4 to 5 megapixel cameras are relatively inexpensive these days—it isn't hard to find one under $300. For even higher resolution, you can venture into the 6 to 8 megapixel category.