A mix engineer, also referred to as "mixing engineer", is a person who, once all instruments, voices, and sounds, etc., have been recorded, creates what is called the final version (also known as "final mix" or "mixdown") of a song, hence the term "mix engineer". He or she 'blends' or 'mixes' the elements of the recorded piece together, to achieve a good balance of the instrumental and vocal volume, as well their frequency contents, while also deciding other properties such as pan positioning, effects, and so on.
A more technical definition: an audio engineer in sound recording, audio editing and sound system

Sound reinforcement system

A sound reinforcement system is the combination of microphones, signal processors, amplifiers, and loudspeakers that makes live or pre-recorded sounds louder and may also distribute those sounds to a larger or more distant audience...

s who balances the relative volume

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

 and frequency

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

 content of a number of sound

Sound

Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...

 sources. Typically, these sound sources are the different musical instrument

Musical instrument

A musical instrument is a device created or adapted for the purpose of making musical sounds. In principle, any object that produces sound can serve as a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates back to the...

s in a band

Band (music)

In music, a musical ensemble or band is a group of musicians that works together to perform music. The following articles concern types of musical bands:* All-female band* Big band* Boy band* Christian band* Church band* Concert band* Cover band...

 or vocalists, the sections of an orchestra

Orchestra

An orchestra is a sizable instrumental ensemble that contains sections of string, brass, woodwind, and percussion instruments. The term orchestra derives from the Greek ορχήστρα, the name for the area in front of an ancient Greek stage reserved for the Greek chorus...

and so on.