"She is not sick today."
Translation:वह आज बीमार नहीं है ।
Only if होना is not the main verb. For example, for "I am not going" (मैं जा नहीं) you can leave off हूँ because जाना is the main verb and हूँ is the auxiliary indicating 1st person present tense. I can't say मैं लड़की नहीं because I need हूँ in there for "I am".
I read somewhere that Hindi verbs function more like adjectives do in English. You inflect the "main verb" for gender the same way you do adjectives -- रहना -> रहता/रहते/रदती, similar to अच्छा/अच्छे/अच्छी. To complete the action, you then conjugate होना as needed to indicate person and tense. If you think of it this way, it's easier to figure out when you're allowed to drop होना.
Does that make sense?