Violet (technical name for purple) is a colour like teal that can be worn by both warm and cool undertone colouring. The difference is that cool undertones can wear all violets, and warm undertones can only successfully wear the warmer end of the violet spectrum.
So which is which?
Consider the Colour Wheel
As violet is in between blue and red on the colour wheel, and blue being the base of cooler undertone colours, it stands to reason that blue-violets are cool and red-violets are warmer.
Cool Violets Have More Blue
So you can see below, the cool violets/purples have more blue in them
Warm Violets Have More Red
Whilst the warmer ones have a pinkish appearance and are more closely related to red than blue.
Picking the Undertone of Purple
Because those with a cool undertone can wear both red and blue easily this is why they can wear all purples (though should stick the the ones that are also related to their best intensity, aka brightness or mutedness of the colour, and value aka lightness or darkness of the colour, as well of course!).