What type of chemical reaction where two compounds react, and the positive ions (cation) and the negative ions (anion) of the two reactants switch places, forming two new compounds or products?

The correct answer is: Double Displacement

The word displace means to push out. A more reactive element can displace a less reactive element out of its compound during a chemical reaction, this reaction is termed as displacement reaction.
Double displacement reaction can be termed as when two displacements take place in a reaction. Also, it can be termed as chemical reaction where two compounds react, and the positive ions (cation) and the negative ions (anion) of the two reactants switch places, forming two new compounds or products
The generic equation for a double replacement reaction:
AB + CD  AD + CB
Here B is replaced by D and D is replaced by B. AD & CB, two completely new compounds are formed.