The Quran is divided into Ayahs or signs, commonly translated as verses. An ayah may consist of a full sentence more than one sentence or part of a sentence. What makes an ayah is revealed knowledge not a decision of the Prophet Muhammad or any scholar. The sequence of appearance of ayat is also revealed knowledge. Ayahs may be Makki or Madani but in the opinion of some scholars, some ayah may consist of a part Madani and another part Makki; an example is the last verse of Surah Muzammil.
The Quran has approximately 6,300 ayat. Due to various conventions used there is a slight discrepancy in counting the total number of ayat. For example, according to one convention Surah Al-Fatiha has seven ayahs that includes Bismillah and another convention the seven ayat excludes Bismillah. One convention includes 113 Bismillah as part of chapters and the other convention excludes it.
A group of ayat has been declared to make up a Surah. Surahs vary in length, shortest ones have only three ayahs and the longest one has 286. There are 114 Surahs in the Quran numbered from 1 to 114. Surah means something divided or walled from both sides.
All Surahs begin with Bismillah except Surah No. 9 called At-Taubah or the Repentance. All Surahs have been divided into Makki and Madani but a Makki Surah may contain a few verses revealed in Madinah and vice versa. There are some differences of opinion among the scholars of the Quran regarding the place of revelation of a few verses and their inclusion into a given Surah. All Surahs have been given names but not titles.
A title is a brief, one word or two words description of the contents but name is not a description. For example, the name John or Yahya does not describe anything about this person but Dr. Yahya has a title Doctor describing his education. Similarly, names of Surahs like Al-Baqarah (the Cow) or Al-Ankaboot (the Spider) are not titles of those Surahs but only names. If they were titles, the Surah would be about the cow and spider, respectively. However, in some rare cases a name of a Surah may also be its title, such as Surah Yusuf.
Referring to the Quranic words and ayat:
The most scientific method is the one given to us by the Prophet Muhammad, that is Surah and verse. Since all Surahs and ayat within them are numbered, the most scientific method is to provide the Surah and ayah number. For example Ayat al-Kursi is 2:255, that is Surah 2 (Al-Baqarah) and ayah 255. Some scholars prefer the reference “Al-Baqarah 255” or Al-Baqarah (2): 255.
The organization of ayat and Surah and their order is revealed by Allah to His Prophet.
Scholars after the Prophet have done additional division of the Quran for the convenience of recitation of the whole Quran in a set period like one week, one month or two months and so on.