From your question I can say that you are following some coaching material. Epsilon means empty string. $L = \{\epsilon\}$ is a language which has empty string in it (here no other strings) and we can draw a DFA for this making it a regular language. A regular language is by hierarchy a CFL, CSL, Recursive and also a recursively enumerable language. This also means epsilon is accepted by a PDA, LBA and also a Turing machine. How a TM accept an empty string is a better question.