InnoDB and MyISAM use B+ and B trees for indexes (InnoDB also has internal hash index).
In both these structures, the depth of the index is an important factor. When looking for an indexed row, a search is made on the index, from root to leaves.
Assuming the index is not in memory, the depth of the index represents the minimal cost (in I/O operation) for an index based lookup. Of course, most of the time we expect large portions of the indexes to be cached in memory. Even so, the depth of the index is an important factor. The deeper the index is, the worse it performs: there are simply more lookups on index nodes.
What affects the depth of an index?
There are quite a few structural issues, but it boils down to two important factors: