A recursively enumerable language is a type of formal language that can be recognized by a Turing machine. This means that there exists a computational process that can list all the valid strings of the language, although it may not halt for invalid strings. In simpler terms, if a string belongs to the language, the machine will eventually confirm it; if it does not, the machine might run indefinitely without providing an answer. These languages are important in the field of theoretical computer science because they help us understand the limits of what can be computed. While all recursively enumerable languages are not necessarily decidable, they provide insight into the complexity of various computational problems and the capabilities of algorithms.