"I will not study today."
From my understanding, adding the は after 今日 would somewhat emphasize the "today" part of the sentence. Sort of like saying, "As for today, I will not study." Maybe you did yesterday and will tomorrow, but you won't today.
Not adding it, which Duo also accepts, would be more of a general statement with no specific emphasis. Just a very general, "Today, I will not study."
It seems a [noun] is combined with a verb [する] and the resulting construction used as a compound verb for the specific activity. There is a funny and interesting Twitter post about it: https://twitter.com/ErikaAlpert/status/1028154746681606144
します is the verb "to do." It can also be attached to the end of certain nouns (like べんきょう - "study") to form what is called a "suru verb," which is basically just a verb form of that noun. So べんきょう + します = べんきょうします = "to do study" / "studying"
います is the verb "to exist" for animate objects, like people. います is also used in many places where English would use "have" as well, like in "I have two brothers."
Works very similar to 'I will not study today'. All you need to do is replace the 'I' with 'She'. But where is the 'I'? In japanese you do not have to reiterate who you are talking about if the subject of the conversation is on them. If no subject has been established yet its safe to infer that the person talking is talking about themselves. For example, if I walked up to a random person and said "ばかです" I haven't established who im calling an idiot, therefore its safe to assume Im calling myself and idiot. In this sentence: "今日はべんきょうしません" no subject is established so you are talking about yourself. If you wanted to make the subject about someone else, lets say 'she', you would just insert that in the appropriate place. Japanese is an SOV language (subject, object, verb) just like how yoda talks if that helps. SO in this case 今日 is the subject, べんきょう is the object, and しません is the verb. In a Japanese sentence names always go first then time (when you are talking about someone in particular). So you would put 'she' right before 'today'. This would give you: かのじょは今日はべんきょうしません。