What is the difference between 把 吧 and 巴? What's the difference between these three and when to use them?
把 is a structural particle.  There are many articles on here and the internet to further understand..it covers too broad a scope for me to answer here.

吧 is usually a modal particle, and comes at the end of a sentence, usually indicating a polite suggestion.  It also is a loan word, which means a word borrowed from English to phonetically represent it's English equivalent.  It is the Chinese loan word for bar.  Another example of this is 沙发 which means sofa, and also can be used to describe the first person who responds to an online post.

巴? According to mdbg.net,
to long for / to wish / to cling to / to stick to / sth that sticks / close to / next to / spread open / informal abbr. for bus 巴士 / bar (unit of pressure) / nominalizing suffix on certain nouns, such as 尾巴, tail 

Ba state during Zhou dynasty (in east of modern Sichuan) / abbr. for east Sichuan or Chongqing / surname Ba / abbr. for Palestine or Palestinian / abbr. for Pakistan