Proteins are built within a cell structure called a ribosome - a protein factory. First the mRNA attaches itself to the ribosome. Then specific adapter molecules, called transfer RNAs (tRNAs), match themselves to their appropriate codons. Each codon has a corresponding tRNA that carries a specific amino acid at one end.

Here is the beginning of a new protein. The first tRNA carrying methionine matched itself to the first codon. Then a second tRNA carrying arginine matched itself with the second codon. A chemical reaction joined one free end (the hook) of the methionine molecule to a free end (the eye) of the arginine molecule, and methionine's tRNA was released.

Now a third tRNA carrying tyrosine has matched  the third codon, and a chemical reaction has joined the free end of the arginine to tyrosine, and arginine's tRNA has been released. The protein now has three amino acids joined together. This process will continue until a "stop" codon is reached.