L1, or first language, is what is referred to the native or indigenous language of the student. It is also referred to as the "natural language", or the "mother tongue". When it comes to language learning and acquisition, the L1 is the language with which the student enters the program and also the language with which the student is most comfortable speaking because of the extensive exposure to it. Therefore, in an ESL setting, the students are expected to revert to their L1 in moments of frustration or in situations of leaning "stress".
L2, or second language, is also known as the "target" language. Any other spoken system learned after the L1, is considered an L2. In the ESL/TESOL field, the L2 is the language that will be acquired through a series of interventions and strategies. The L2 can either be a "second" language or a "foreign" language; there is a difference between ESL and EFL in terms of how the student will use the language after it is learned. There is also EIL (English as an International Language) and there is also English for Specific Purpose learning.
Do not think, however, that L1 and L2 apply only to English; it is often the case that the terminology is used in TEFL settings only because TEFL programs are more extensive and widely-used throughout the world. Realistically, you can apply the terminology of L1 and L2 to any setting that involves learning two different language systems. Some linguists have gone as far as re-labeling the L1/L2 as L1 and FL2 (foreign language 2) but that is not a common nor productive practice.