Teus is the informal version. You would use it when you use tu, and that is used for friends, family, etc. Suas is the formal version. It is used with você. So, use it when referencing an elder or someone of authority. Easy way to remember it: a ma'am or sir is a você, a person you are familiar with is a tu
Maybe Duolingo is just being didactic, but this sentence is not common Brazilian Portuguese. você/seu is used all over Brazil, whereas tu/teu is used only in a few regions and Portugal, and yet they also use and understand você/seu. seu can also mean his/her (since it's the proper pronoum for the 3rd person) but in Brazil it's used only in formal/written language with its meaning.
o meu carro = my car
o seu carro = your car
o carro dele/dela = his/her car
o nosso carro = our car
o carro de vocês = your car (plural)
o carro deles/delas = their car
Other uses are:
o teu carro = your car [used (alongside seu) only in a few regions]
o vosso carro = your car (singular/plural. archaic (used in bibles, prayers...)
o seu carro = his/her car in formal/written language
The "s" here is because of the plural. The Portuguese possessives must agree with the noun (cavalos) they point to.
Teu cavalo (your horse)
Teus cavalos (your horses)
Tua égua (your female horse - mare)
Tuas éguas (your female horses - mares)
Portuguese doesn't have different words for "your" and "yours"
This is your horse = este é teu cavalo
This horse is yours = este cavalo é teu
Those are your horses = aqueles são teus cavalos
Those horses are yours = aqueles cavalos são teus.