In a biological context, translation refers to the synthesis of proteins encoded by mRNA (messenger RNA). Translation is a fundamental activity of all cells so it must have evolved very early in the history of life. It is remarkable that nature engineered (rather most scientists assume it did) such a complex process, which has remained unaltered for billions of years and has allowed life to flourish on the Earth. One of the most exciting frontiers of science is the understanding of how complex processes such as translation evolved from the chemically-simple, inanimate Earth.
Translation occurs after (or during in the case of prokaryotes) transcription of DNA to mRNA. In eukaryotic cells, before translation the introns (non-coding portions of the mRNA transcripts of genes) are excised from the mRNA and the exons (coding portions of the transcripts) are stitched back together. The excision and splicing process is known as mRNA processing. The mRNA strands eventually make their way to ribosomes where they will be translated into proteins.
Cells continuously produce molecules of tRNA (transfer RNA) bound to amino acids. The binding is catalyzed by 20 enzymes known as activating enzymes. The enzymes bind a particular amino acid to a tRNA molecule with a corresponding three-nucleotide sequence (called an anticodon) that will later bind to a three-nucleotide sequence (called a codon) of mRNA. Since four nucleotides--adenine, guanine, cytosine, and uracil--are used, there are 43 = 64 possible anticodons. This implies that some amino acids correspond to more than one anticodon. For example, the amino acid leucine can be binded to tRNA with anticodon CUX (X is any nucleotide), while tryptophan binds only to tRNA with the anticodon UGG.
Translation of mRNA into protein
Translation occurs on ribosomes, organelles made of protein and yet another kind of RNA--ribosomal RNA (rRNA). I will not discuss the physical structure of ribosomes or the details of how they bind mRNA to tRNA since the ribosomes are difficult to visualize and the biochemistry is over my head. The first step of translation is the binding of a tRNA molecule (known as met tRNA) carrying the amino acid methionine to a ribosome. The anticodon (UAC) of met tRNA corresponds to the start codon (AUG) of mRNA. An mRNA strand binds to the ribosome and tRNA such that the start codon and met tRNA anticodon bind. The second mRNA codon attracts a corresponding tRNA anticodon, and they bind with the help of the ribosome. The two amino acids carried by the tRNA molecules form peptide bonds, resulting in the release of the met tRNA molecule.
At this point the protein consists of two amino acids--methionine and another, depending on the second codon. The mRNA moves three nucleotides relative to the ribosome (in a process known as elongation), leaving the third codon in a position where the ribosome can help it bind to its anticodon. This process repeats tens to hundreds of times resulting in a long amino acid chain, or protein. Eventually the mRNA reaches a stop codon (UAA for instance). No tRNA binds to a stop codon, but cleaving enzymes do bind to it and release the polypeptide chain. Thus the mRNA has been translated into a protein.