Digital ICs use logic gates, which work only with values of ones and zeros. A low signal sent to to a component on a digital IC will result in a value of 0, while a high signal creates a value of 1. Digital ICs are the kind you will usually find in computers, networking equipment, and most consumer electronics.
Analog, or linear ICs work with continuous values. This means a component on a linear IC can take a value of any kind and output another value. The term "linear" is used since the output value is a linear function of the input. For example, a component on a linear IC may multiple an incoming value by a factor of 2.5 and output the result. Linear ICs are typically used in audio and radio frequency amplification.