“affaires” in the meaning of “thing” is mainly (but not always) used with possessive pronoun (i don’t know if this is the right english term, it’s literal translation from french) and plural like “tes” (your), “mes” (my)… “mes affaires” can be translated as “my stuff”. Used like this it is also uncountable. “As-tu vu mes affaires ?” = “Did you seen my stuff ?”, “Prends les affaires de Tom.” = “Take Tom’s stuff.”

Used as “les affaires” with no complement means “business” (uncountable again) : “les affaires sont les affaires” = “business is business”

Finally used with singular it is countable and is closer to “case”. “Cette affaire m’ennuie.” = “This case annoys me.” (this particular meaning is likely to be linked in a way with the english word “affair”).

“objet” is a very generic word that is very close to “thing” and “object”. You won’t use it with possessive. “mes objets” sounds weird.