I hope you find these descriptions useful. I will try to describe them in simple, practical terms.
ESL - English as a Second Language (or as I liked to call it - English as a Subsequent language...as many students have English as a third or fourth language). This is the term that the ESL document - the ESL Scales - uses and it is also the term the old and new NSW English syllabus refers to.
EAL/D - English as an Additional Language/Dialect. This is the term used on the new ACARA document that describes the new phases used to gain ESL funding in the Public School System. Personally I think it is a better fit for the term ESL and hope it is eventually used to rename the ESL Scales!
NESB - Non English Speaking Background. You may hear students described as NESB. This means they generally have no or little English spoken at home, and it is often applied to newly-arrived Phase 1 students when they first start learning English.
LBOTE - Language Background Other Than English - This term applies to students who have a language background other than English. This means the student may speak English at school and a different language at home. It also may mean they speak English at home in combination with another language. Often teachers are surprised to find they have LBOTE students in their class, who may speak a different language with extended family or who are exposed to a different language by their extended family. Some LBOTE students understand their home language but don't speak it fluently, some speak English most of the time but might, for example, communicate with grandparents in a different language.