Beats audio means the following hardware-wise:
1. Redesigned headphone jack – reduces ground noise
This means that the jack components are made from plastic instead of metal so that the possible ground noise is minimized. This is the right way to do the headphone jack anyway, so I don't think you'll find any phone out there except some cheap Chinese Android that might have this oversight.
2. Discrete headphone amp – more powerful with better stereo separation
This means just that there's a separate amplifier circuit for the headphones that will increase the output signal level that comes from the audio circuit. The output level of that should be about as much as with your laptop, since most laptops usually have also headphone amplifiers built in. It should not affect the quality of the sound, but if you have a low sensitivity headphones you might hear more and that way perceive the audio as of better quality.
There's also a ground cable in between the stereo cables from the jack to the amplifier circuit and better insulation of the cables themselves, so that there is less possible interference. It should be noted that if you use crappy cheap headphones that don't have any similar considerations in their build quality, you won't get the advantages this kind of build allows.
3. Dedicated audio island – isolates audio components on the circuit board
The dedicated audio island is exactly what they say it is — simply isolating audio components found on the circuit board so that there's no interference from other components. As it should be done, Beats Audio or not.