Mipmaps are basically smaller versions of the original texture:
So, depending on the distance/how much of the screen your model covers a different mip-map is chosen. If you are very close the largest version (the original size) of the texture will be used. You'll be able to see details and stuff. As you move further away, a smaller version of the texture will be used (a mipmap). This will sort of 'naturally blur' the texture to better account for the distance you are viewing it. It's also nice for performance, as objects further away use a smaller texture (less memory used).
Not using mipmaps can produce weird results:
Left = no mip maps
Right = mip maps
You can also get a grainy texture if not using mipmaps (as the blurring doesn't occur):
http://i.imgur.com/L8TtEFQ.jpg
(The image to the furthest left is the same in both top and bottom, and it's the image shown through the entire series of images at the bottom.)
The images at the bottom has no mip-mapping on the diffuse map - and you can see how they do not really blend well, but rather turn out quite grainy.