ESL students are people learning English as a second language, that is, as another language after their mother tongue or primary language.
ESL programs are mainly for people living in an English speaking country, for example many US citizens speak Spanish and learn English as a Second language to live and work in the USA. Likewise immigrants to a country like the UK will need to learn English to live there successfully and this will be English as a Second Language.
Teachers who teach ESL classes are defined as TESL teachers as they Teach English as a Second Language, which is different from teaching English as a native language. To become a TESL teacher you need specific teacher qualifications.
Compare ESL to EFL which is English as a Foreign Language. This is taught to students who are living in a non-English speaking country and wish to learn English for business or pleasure, etc.
EFL - English as a Foreign Language - all about EFL (compared to ESL)
TEFL - Teaching English as a Foreign Language - all about TEFL
TESL - Teaching English as a Second Language - all about TESL