Synch HO : when the serving cell and target cell are sharing the same clock (they belgon to same BTS), or they are synch thanks to external device (GPS).
Async HO : all other cases.
Synch HO does not require the target cell and the MS to synch during the HO procedure. Therefore, they are faster to be finished (much faster), the loss of voice during HO is not perceptible.
There are only two types of HO : Synch or Asynch.
Then there are big families : Internal HO (within same BSC) or External (btw different BSC), or Intracell.