ESL stands for English as a Second Language, whereas ELL stands for English Language Learner. Therefore, ESL is the class and ELL is the student in the class.
ESL classes are for students who do not speak English as their native language. They may be students who moved to the United States from a different country, or they may be students whose families speak a language other than English in the home. Laws regarding providing ESL instruction vary from state to state and from district to district in many cases.
ESL instruction more commonly goes by ELD now, or English Language Development. Parents enrolling their children in public school typically answer a questionnaire related to the home language. If a language other than English is indicated, the child takes a test. If ELD instruction is warranted, the school provides it. In addition to developing the child's English language skills, teachers focus on concepts standard in a Language Arts classroom. In most cases, all teachers must provide some level of support for the ELL, or the student who is learning English.
ESL is still the term used for classes in private language schools. The students in these schools might be immigrants who need to learn English, or they might be students who come for a specified amount of time to learn English. Instruction in these classrooms looks different from the ELD classroom, as it is focused only on English. These students are generally called ESL students, not ELLs.