(A quick answer to a question that will be likely closed being not well focused on the physics of all this)
First: the ribosome translates mRNA into the chain of amino acids that eventually will become a protein. The ribosome is a huge complex, made of both proteins and RNA. RNA can have enzymatic activity like a protein: the ribosome is a Ribozyme.
This chicken or the egg problem has puzzled biologists, chemists and biophysicists for a while. The structure itself of the ribosome was the "smocking gun" for the model we have today: the RNA world. In few words: because RNA can act as a protein and it is so similar to DNA, probably our DNA-world (i.e. information stored on the DNA, work done by proteins, RNA between the two) emerged from a fully RNA-world, where RNA was responsible for both protein activity and information storage. How in detail all this happened by evolution, is probably the subject of several nobel prizes to come.
