The division of the bible into chapters was probably done in the 13th century by Stephen Langton, a Dominican. The books were divided into chapters, and then each chapter into other divisions (either 4 or 7) and lettered (A,B...G). There were other systems available, but Langton's one stayed with us.

The division of verses has a more complicated history. Hand-written manuscripts were done by scribes who often charged per line, so the number of lines in a book was important to know (this is called stichometry). Biblical books were initially 'counted' by lines but this isn't very helpful if you need to reference something in the middle of Jeremiah. The Masoretes counted the number of verses in their bible which was another form of stichometry, but this still isn't the verses that we have now. The habit of actual 'numbers' attached to the verses start in 1448 with Rabbi Isaac Nathan's Hebrew concordance. This was rather quickly adopted by Henricus Stephanus (Henry Estienne) who thankfully employed Arabic numerals rather than Hebrew ones in his 1509 edition of the Psalms, and his son, Robertus Stephanus (Robert Estienne) did the NT in that fashion in 1551, and then a few years later did a completely versified Vulgate in 1555, apparently, much like Nathan, to help with doing his concordance. His son, (confusingly called Henry) said that Robert's contemporaries ridiculed him and said it would bring him nothing but derision, but the numbering system was almost instantly adopted by everyone, sidelining other systems such as Sanctes Paginius' system.

Estienne's system was adopted by the Geneva bible, which was the most popular English-language bible (about 161 editions) and that together with Theodore Beza's adoption of the numbering into his Greek NT editions helped cement the concept, and as the KJV stole quite unashamedly from the Geneva bible, all English-language bibles ended up using Estienne's numbers.