If I say:
My sister can drive the car today
Which is the subject: sister or car? Also, what do you call the one which is not the subject?
Subjects in English generally come before the verb. The verb can have "other things" near it, words like have, can, should, was, etc. The subject can be things other than a noun, such as car, tree, or Bob. They can be words like "skiing" "going" or "eating." The basic sentence pattern in English is Subject + Verb + Object. The object is usually the thing acted upon by the subject. Sentences do not always have direct objects or even subjects. "I am eating right now." does not have an object. "right now." is not something that you can eat.
In "Bob eats a corn dog." "corn dog" is the direct object. In "Bob gave Cindy a corn dog." "corn dog" is still the object. "Cindy" is called an indirect object. She received the object. We know this because we can put "to" after Cindy as in "Bob gave (to) Cindy a corn dog." That's a bit awkward sounding so we reverse the sentence, "Bob gave a corn dog to Cindy."
I go into a bit of extra detail for you because indirect and direct objects are things which confuse learners or native speakers when learning grammar. I sometimes have to stop and "figure things out" in a sentence, so don't feel bad.
In your sentence, "My sister can drive the car today." "My sister" is the complete subject, "my" is a possessive pronoun, "can drive" is the complete verb. "Can" is a modal that tells more info about drive. "Can" is sometimes referred to as an auxiliary or helping verb. "The" is a definite article (telling which car), "car" is the direct object, and "today" is a word that can be either a noun ("Today is the beginning of the rest of my life.") or, as in your example, an adverb telling more about drive. Adverbs can sometimes answer "how?" "when?" and "why?" questions. The word "today" could be confusing...maybe "today" is "the other thing." It is not.
Hope this helps.