Surat al-Tawba is the only surah of the Qur'an that does not begin with the Basmala, the words "In the Name of Allah, the Most Compassionate, the Most Merciful." It was one of the last surahs to be revealed, in the 9th year after the Hijrah. Two explanations for the missing Basmala are reported from the sahaba, the companions of the Prophet ﷺ.
The first comes from Uthman ibn Affan RA, who compiled the written copy of the Qur'an (the mus'haf). According to a hadith in Sunan al-Tirmidhi, he was asked why he had placed al-Tawba straight after Surat al-Anfal. He explained that al-Tawba came very late, and the sahaba were not sure where it belonged. They put it after al-Anfal because both surahs deal with warfare. Since they did not know whether the two were one surah or two, they did not write a Basmala between them.
Because of this, some of the next generation (the tabi'un) counted al-Anfal and al-Tawba as a single surah, giving 113 surahs in total. Hardly anyone else agreed. The great majority of the ummah, and the later consensus (ijma), holds that they are two separate surahs.
The second explanation comes from Ali ibn Abi Talib RA. One of his sons asked him why al-Tawba has no Basmala. Ali RA answered that this is a surah in which Allah cuts off His ties with the pagans, so it is not fitting to begin it with His mercy. The surah's very first word is bara'ah, which means a complete cutting off. In another narration, Ali RA explained that the Basmala is what one writes at the start of a contract of protection, a promise of mercy. Al-Tawba is a surah of dissociation, so it should not open with "the Most Compassionate, the Most Merciful."
The missing Basmala also raises the question of who decided the order of the surahs. Everyone agrees that the order of the verses inside each surah came from Allah through the Prophet ﷺ. The order of the surahs themselves seems to have been set by the sahaba. Uthman RA's own answer suggests this, and so does the fact that companions who kept personal copies arranged the surahs differently. Once Uthman RA standardized the mus'haf, the ummah agreed on his order, and following it became binding.
O Amir al-Mu'minin, why did you put Surat al-Tawba after Surat al-Anfal?
Al-Tawba was one of the last surahs revealed, and the matter was unclear to us where it should go. Therefore, we put it with Anfal because the content is the same. And we did not put a Basmala not knowing if the two Surahs are connected or not.
Why is there no Basmala in Surat al-Tawba?
This is a Surah where Allah cuts off His ties with the pagans; so it is not befitting that He begins it with His mercy.