Take an example; "'Grass is green' is true if and only if grass is green". In this sentence the subject part, which is in a single quote, is treated as a meta-language because it is just the concatenation of words. It does not convey any meaning. It expresses meaning when the predicate part affirms the facts by examining their availability in the empirical world. The predicate part grass is green which is free from quote conveys that there are objects called grass and their colour is green. This is called the object language as it deals with objects of the world. But the subject part of the sentence, which is in a single quote, is treated as meta-language because it does not convey any meaning on its own. It needs the help of object language for its explanation.
The third variety of language is known as 'logically perfect language'. It is a unique language which is free from all sorts of logical errors, those we often encounter in the ordinary language usage. Its nature is exhibited as well structured, grammatically correct, referent assigner, and meaning conveyor. The meaning it conveys is about the worldly affairs- it may be about object or concept of the world. Sentences belonging to this language offer both sense (meaning) and reference. It is propounded by Frege, which he calls Begriffsschrift.
According to Frege, natural language (ordinary language) is not a well-defined and perfect language, because the defects lie in the structure of the sentence. For example, let's consider a sentence, "Practicing austerity is a good habit". In this sentence, how do we refer to the words 'practice' and 'austerity' and further, what will we refer to when we say that something is a good habit? In this way ambiguity arises in the natural language. In order to overcome these lacunas, specially relating to the reference failure, Frege suggests that we need a formal and logically perfect language. A language where in every sentence is endowed with both 'sense' and 'reference'. As a result, all the sentences can be judged as either true or false.