I'm trying to understand the concept of semiconductors, as far as I know whenever an electron moves it leaves a hole, so how can we say (for example) that n-type semiconductor has a majority of \$e^-\$, won't every electron let a hole?
What you are confusing is the application/use of semiconductors with the manufacture of the semiconductors.
The of type semiconductor is determined by the type of "impurities" added by the manufacturer to the silicon. If the impurity has more electrons than it needs, it is an n-type, if it has fewer electrons than it needs, it is a p-type. These materials, when vapor deposited "next to each other", form a p-n junction (band).
Now, when you apply a forward bias the junction, the electrons of the n-type material will be pushed into the "holes" of the p-type material leaving holes on the n-type side of the junction (which will be filled with new electrons coming from the negative side of the supply). The electrons that went into the p-type are attracted to the positive side of the supply leaving holes on the p-type side of the junction.
So it is clear that a "hole" is created when an electron moves, but the electron would not be able to move, in the first place, if there were no "holes," to move into!