Generally, the type of visa that you may use mostly depends on the duration of your visit. However, a tourist or business visa cannot be used if:
you visit Russia in order to adopt a child (a specific adoption visa will be required)
you want to send a child to Russia with his/her Russian mother (a homestay visa will be required)
you want to stay in Russia more than 90 days (a student or work visa will be required)
In these cases invitations must be processed by the persons or entities you really plan to visit.