As with all classical books, significant editing is required to publish them in modern format. Hadith books can be organized in many different ways - the most popular being by topic (Jami') and by narrator (Musnad). Sahih al-Bukhari is a Jami'.
In Imam al-Bukhari's time, hadith were known by their narrators and subject matter. They weren't referred to with numbers, the way we do now. It was the same with ayaat of the Qur'an - they didn't go around saying things like "fifty one, fifty six" to refer to ayaat in the Qur'an. Rather, people were familiar enough with the Qur'an to navigate to wherever it was that someone was referring to. Additionally, the culture we have today of plucking out and citing singular hadith in support of a matter was simply not the norm in those days, nor is it the norm in scholarly circles today. For this reason, numbering the hadith was not crucial, and the collector (Imam al-Bukhari) did not number the hadith in his book.
The most popular edition of Sahih al-Bukhari that we have today is the one that is transmitted through one of his last students, Muhammad ibn Yusuf al-Firabri. Others students of al-Bukhari transmitted it also. This is the basis of all printed editions in the world today. Muhammad Fuad Abd al-Baqi went through the collection and assigned numbers to each hadith (this is not as simple as it sounds). His numbering is the most widely used today to refer to Sahih al-Bukhari, although alternative numbering systems exist as well.
Imam al-Bukhari divided his collection into books, and then chapters (*bab*s). On average, a book might contain 70-80 hadith, and a chapter within a book 2-3 hadith. There are 97 books in Sahih al-Bukhari according to the Abd al-Baqi numbering containing 7563 hadith. Each book deals with a coherent topic. Hadith are repeated in different chapters and books on average three times, although there are hadith that appear once, and others that appear over 10 times.
The way to understand hadith in Sahih al-Bukhari is to first see which book it occurs under, then which chapter it occurs under. Sometimes a chapter heading is accompanied by a hanging narration (mu'allaqat) in which Imam al-Bukhari presents a quote from the Qur'an or the Prophet (saws) or the Companions (raa) or a scholar from the Salaf without the full chain of narration. These are not considered independent Sahih hadith, but rather part of the chapter heading. These headings are crucial in understanding the context of the hadith and what is meant by it. Once these have been read, one can proceed to read the hadith.
There are many ways of referring to a hadith in Sahih al-Bukhari. The most foolproof and consistent one is to specify the name of the book, the name of the chapter, and either the name of the narrator or the position of the hadith in that chapter. Another way is to specify the hadith number as assigned by Abd al-Baqi (a number between 1 and 7563). Other methods are not as portable.
The most famous numbering system on the Internet today for translations of Sahih al-Bukhari was popularized by the USC MSA hadith website. It usually consists of a volume number, book number, and hadith number. This numbering is entirely arbitrary unfortunately and does not correspond to any printed edition. It has a number of mistakes and should be phased out.