"I am not busy this week."
～ません is mainly used in written Japanese; ～ないです appears more often in spoken Japanese.
They're roughly equivalent, but there are some differences: https://japanese.stackexchange.com/q/2574/
According to the source cited by one answer:
masenhas a strong sense of formality, and often expresses the speaker's firm denial. Also, it is often used to add a sense of assertion or in emphatic constructions. This is thought to arise from the influence of the linguistic form (where the order is polite
n), in which the intent of negation appears stronger than the politeness. From this the speaker can draw the listener's attention to the negating function. If we assume that expressions which appear nearer to the end of the sentence leave stronger impressions, the negating function of
masenleaves a strong impression.
nai desuis often used in spoken Japanese. As indicated by Noda (2004), the effects of this construction are such that it is often used in concert with sentence-ending particles, and that it is easily joined with the
(verb) + shiteiruform. This is thought to arise from the influence of the linguistic form (where the order is negation
desu), in which the level of politeness appears stronger than the negating function. From this the speaker can draw the listener's attention to the level of politeness. If we assume that expressions which appear nearer to the end of the sentence leave stronger impressions, the politeness of
nai desuleaves a stronger impression. Also, it is further thought that this may have the effect of softening the speaker's strong assertion of a negative expression.
～ません and ～ないです are semantically equivalent, but ～ないです is softer and less insistent. If you need to give a firm denial with no wiggle room, go with ～ません. But since this level of bluntness can be inappropriate in some situations, ～ないです is there if you need it.
Japanese verbal variations are extensive, and sometimes while learning it can serve more in confusion when swapping between formal and informal literally from one test window to the next. Typical real life scenarios would see ourself and our interlocutor in the one mode for the conversation duration. It would be nice if occasionally there were categories here and there through the course which concentrated on formal only and informal only to aid knowlede consolidation. I feel particularly for those who are here to start from scratch where they will not have any kind of rule book knowledge available which is in the "traditional" courses. Like many here, I have studied elsewhere and know without the previously gained grammar rules, I would be making mistakes far more and more often.
You can down vote me on this all you wish but during my Japanese classes it was strongly emphasized that nobody in Japan forms negation of -na adjectives using -ku + "arimasen". the question is if isogashii is a -na adjective or an -i adjective?? You just do -nai + "desu" all time, as mentioned in the comments here (that it is common for spoken Japanese) and the same goes for the past tense, isogashi-katta desu and isogashi-kunakatta desu. Even though the 'arimasen' version is correct, if you speak like that you make it clear you are a learner / foreigner / not fluent in Japanese.
Maybe it changes and it gets more complicated as you get more advanced but I will stick with my native speakers who taught me how to speak, for now