This means that a signal present at the mixer input separated from the desired signal by twice the IF frequency will be indistinguishable from the desired signal when it gets to the IF. Any signal at this "image" frequency must be removed by filtering before the input of the mixer, or it will create an unresolvable interferer in the IF passband. Techniques have been developed to reduce this problem -- special mixers known as image-reject or single-sideband mixers can reduce the problem significantly, but they require duplicating most of the mixer circuitry, and consume more power than a normal mixer.
Demodulation in most modern wireless systems is accomplished by decomposing the signal into its I/Q (In-phase and quadrature) components. These I/Q signals are generally converted from analog to digital and digital signal processing is used to extract the modulation (and correct for channel imperfections, multi-path, fading, etc.). The I/Q demodulation is performed on the IF signal in a modern superhet receiver, with the demodulating LO usually at a fixed frequency.