Here is the full answer to your question.
It depends.
The valid language test that is less than 2 years old is mandatory for the primary applicant when you are using EE.
Following language tests are accepted:
IELTS General
CELPIP
TEF or TEF Canada (not TEF Quebec)
Dependant applicant (your spouse and children) are not required to have one. However it will give you more points for spouse (since you get less points for the main applicant if you have a spouse).
But with that said, application that is paper based (not EE) does not have language test as mandatory (at least not the federal part).
If you use one of the paper based PNP (provincial nominee) programs, it depends on the requirement of the province and program itself.
If you use other streams (sponsorship for spouse and children, refugee claim, sponsorships for parents and grand parents, Atlantic program or live in caregivers), language test is not required (unless stated as a requirement for the specific stream).
Language test requirement for EE, is for any foreigner regardless of his or her citizenship or secondary or post secondary studies in one of the official languages (be it abroad or in Canada).
Also the minimum requirement for the score in order to be accepted depends on program (EE is CBL 7 for FSW and CEC applicants and CBL 6 for FST applicants for the first language and CBL 4 for the second language; PNP have different requirements ranging from CBL 4 to CBL 7 as a minimum requirement (where is such requirement).
So this is the full answer to your questions.