"I am not busy today."
Since all the replies have got it wrong, I'll help you out instead ;-)
ありません explicitly states that something inanimate exists, you just said that today exists and I don't know what "busy" is doing there.
It should most definitely be either ないです or でわありません.
ない is used as the negative counterpart of ある.
ありません is the negative of あります.
あります ( ある→あり + ます)
ありません ( ある→あり + ます→ませ + ぬ→ん)
That should be ではありません.
ではない (plain) → ではありません (polite)
However, ではない (or じゃない) and ではありません (or じゃありません) are instead used after nouns and for な-adjectives. These aren't used with い-adjectives like 忙しい...
くない (plain) → くありません (polite)
忙しくない (plain) has the same meaning as 忙しくありません (polite).
です thrown on the end of 忙しくない (plain) → 忙しくないです (polite).
People say both of these forms all the time. As others also point out in this discussion, ないです and ありません basically equivalent. ありません is the polite form of ない, but adding です after ない is another way to elevate the plain negative to the polite level.
今日は忙しくありません is grammatically correct and a more proper way of speaking or writing. Using 忙しくない is more casual. Both answers should be accepted as correct.
You can see that 忙しくありません is used in many examples on Weblio, a Japan-based Japanese-English dictionary/translation website: http://ejje.weblio.jp/sentence/content/not+busy
I feel like tgis same exact form is the "correct" format on another question that replaces "today" with "this week"
I think your answer is correct.
Yes, yes you could, but remember that being formal is the standard when talking to japanese people です
いそがしく is the conjugated form of いそがしいいそがしい. It's used for things like negative forms (as it's used here) and てて forms among others. (For future learning, all い adjectives conjugate this way: you replace the い at the end of the adjective with く and then add whatever form you're conjugating it to at the end.)
I know that relative time expressions don't take に, but why is it that only some use は (like here)?
に is used as a pointer, more like "in" or "at" a certain, more specific, place or time. は seems to be not only better for general time periods: "as for last month" etc, it is used in this to emphasise it being a negative.
It marks 今日は忙しくないです wrong and gives the following as a correct answer: 今日はいそがしくないです the only difference being that the latter lacks the kanji for busy
Check the top comment. It is equivalent and two ways of saying the same thing.