New tyres can be very expensive, particularly if all four need replacing, therefore many people turn to buying part worn ones as a cheaper alternative. Part worn tyres generally come from cars which have been written off due to faults and accidents whereby the tyres were not affected.
Part worn tyres must be legal in terms of tread depth and condition in order to be resold; if they are cut or damaged then they will be unsafe to sell. Be aware that when you are purchasing these cheaper tyres, they will not have the same length of life left as new ones will, so perhaps can be a false economy in the long run. However, as a cheap immediate fix, they work perfectly.
The tread on the tyres must be 1.6mm deep at the very least to be road legal; you will not be able to regulate the depth of tread on your part worn tyres when you buy them so a set of 4 could have different tread depths and therefore last differing amounts of time.