Gelish is a Gel Polish, basically a thinned down Gel which can be applied by brush like polish. The nail plate requires to have the shine removed before application to help with adhesion. Also you require to buff the surface of the Top Coat prior to removal.
Loads of colours available and new ones launched several times a year.
Shellac is known as a power polish but that is a marketing term, it is technically a type of gel polish..but more like polish than Gel. Also applied by brush like a polish, no removing shine,or buffing of the nail plate is required with this system. Removal is quicker and requires no buffing of the topcoat. Great selection of Classic colours but not as much choice as Gelish.
There are literally hundreds of threads on here about both....try searching them out In the search facility on here.