A mixer, sound mixer or mixing desk as it is sometimes called is a device that takes multiple sources of sound and allows a sound engineer to balance them against each other and apply equalisation or audio effects to the sound source. Each sound source must have its own volume level compared and balanced against others for each instrument to achieve an equilibrium that is considered to be acoustically harmonious.
A mixing desk does this by presenting each instrument with its own channel input, thereby separating each sound source from the others and giving the sound engineer total control over its volume, EQ and effects.
An amplifier is an electronic audio device that can amplify low powered audio signals and boost them to a level that is suitable to drive a collection of loudspeakers and thereby achieve audible output from the speakers.
Although sound mixers and amplifiers are generally separate parts of a PA system, there is a hybrid product known as a mixer amp where the amplifier and mixer are integrated within the same chassis.