So, what is it that a tone-deaf person isn't perceiving correctly? The answer is pitch. Pitch is the highness or lowness of a sound. If you were to play the first key on a piano keyboard, you would hear a low note, or low pitch. Playing the last key on the keyboard would produce a high note, or high pitch. Why do these keys sound different? The keys are attached to taut strings of different lengths inside the piano. When these strings are plucked (as with any stringed instrument), they vibrate. A vibration extends the entire length of a string. So an amount of force exerted on a longer string will result in a slower vibration (lower note), and that same force on a shorter string will produce a faster vibration (higher note). This vibration transfers to the air around it, ultimately reaching your ears, which begins the brain's processing of a sound.
Through thousands of years of trial and error, humans have selected certain vibrations to be musical notes, since they sound better than other vibrations. The A note of a violin consists of 440 vibrations per second, and if there are a few less or more, it will sound "off" to a majority of people. (Pitch for a particular note can change, however. For instance, different orchestras sometimes assign different frequencies for a given pitch, and a note played at sea level may play at a different frequency than the same note at high elevations in order to produce the same-sounding note.)
Most people can't name a note after they've heard it -- you don't hear someone sing and think, "Oh, that's an E." But if you heard another note, you'd be able to tell which note was higher and which was lower. Tone-deaf people, however, can't differentiate between two different notes, unless perhaps they're radically different.
Now that we know what tone deafness is, we'll take a look at what may be causing it in the next section.