The standard mosaic tiles on walls can use a 1/8-inch trowel to apply sufficient mastic or thinset for the tile to bond with. Natural stone mosaics can sometimes get away with 1/8-inch, but more often they need a 1/4-inch trowel instead. Ceramic or porcelain tile in 4-inch pieces can be installed with a 1/8-inch trowel, while their natural stone counterparts require a 1/4-inch notch.
Manmade tiles such as porcelain, corks, glass and ceramics can be installed with a 1/4-inch trowel if they range between 4 inches and 12 inches. Any manufactured natural stones like polished marble and granite or honed travertine will also use the same size notch. Slate, tumbled marble and other raw stones may require a larger notch; alternatively, spread the thinset on the tile as well as the wall to make up for skinnier pieces in a batch.
Most ceramics and porcelains can use a 3/8-inch notch for everything from 12 inches up to 18 inches. The rule for natural stones applies as with other sizes of tile; they require larger notches for working with the raw versions with their variations in size. Anything beyond 18 inches will require a 1/2-inch notch for most applications and 3/4-inch for the largest or least uniform pieces.
For the most part, raw natural stones will require a larger notch because of the variances in size on the backs of the tiles. However, when you use a larger trowel you naturally apply more thinset, and the variants with tiles like slate may require that you remove thinset from under the corner of one edge but fill up the other edge with additional material.