What type of reaction is the generic equation a + bc → ac + b?

This is a single displacement reaction. Where a single element can displace an element in another compound and form a new compound.

In the reaction given;

a + bc - - > ac + b

bc is a compound, if a is more reactive than b, it can take the place of b and form a new compound ac.

Since only one displacement reaction is taking place its called a single displacement reaction.