Bengali script, sometimes called Bangla, evolved gradually from Nagari, which makes giving it a starting point difficult. Certainly Bengali was a separate script by 1778, when the first metal type was cut for it, but the date sources say it diverged from Nagari ranges from 1000 to 1200.
Worse, some people define an intermediate script “Eastern Nagari” between Bengali and Devanagari; others say what I am calling “Bengali script” is “Eastern Nagari script”. (I suspect that is for political reasons, as there are two minor variants of this script used to capture spoken languages that are not Bengali. English is written in a script that type (font) geeks call “Latin script”; people don’t mind much. I bet there would be serious objections if that script was called “Italian script” instead.)
Like all of the Brahmi-derived scripts, Bengali is an abugida, where most characters are syllables with a consonants plus an implied vowel. Unlike most of the other Brahmi-derived scripts, Bengali script’s implied vowel is pronounced as “oh”, while most other Brahmi-derived scripts pronounce it somewhere between an “ah” and “uh”. This would be similar to if, say, Spanish pronounced the letter “A” as “ah” and Swedish pronounced is as “oh”.
Note that even though the implied vowel is pronounced as an “oh”, the characters are described or named in many places with the “a” vowel. For example, Omniglot describes the character at the top of this post as the “cha” character. This makes a certain sense: it is related to the character that all the other Brahmic scripts pronounce as “ah”/”uh”. This is similar to how we English-speakers describe a descendant of “a” as “a-umlaut” even though Gremans pronounce a-umlaut close to the way Spanish speakers pronounce the letter “e”.