This is a very complex and intricate process, facilitated by many proteins. The DNA is separated into two strands (by helicase), and then one of the strands is "translated" (transcribed) into the equivalent RNA. This is called mRNA (messenger RNA). Then the RNA undergoes some processing, and is sent out of the nucleus to a ribosome. The ribosome then translates the sequence of mRNA into a protein, by attatching successive amino acids into a chain. The chain folds up to become a finished protein.
Protein synthesis can be broken down into two sections: Transcription and Translation.
Transcription is the process by which mRNA is created based on DNA. The enzyme RNA polymerase binds to the DNA at the beginning of the gene sequence to be copied. It then "unzips" the region, breaking the bonds between DNA base pairs.
RNA nucleotides then align themselves opposite one of the stands, with each nucleotide corresponding to one of the bases on the DNA. Guanine-Cytosine, Thymine-Adenine, Adenine-Uracil.
The RNA polymerase then joins the RNA nucleotides together to form a single strand of mRNA next to the DNA strand, which then rejoins the rest of the double helix.
Translation occurs outside the nucleus of the cell. Amino acids are attached to transfer RNA molecules. Each tRNA molecule carries one amino acid, and has one set of three bases called the anticodon. The tRNA molecules and the mRNA molecule join to a ribosome, where the amino acids are bonded together in the order of the bases on the mRNA. Peptide bonds are formed between them, and when the end of the mRNA molecule is reached, the polypeptide detaches.
It is better to refer to the product of this process as a polypeptide, rather than a protein, as some proteins e.g. Haemoglobin, are in fact made from two polypeptides, and an inorganic molecule or ion.
DNA is transcribed into mRNA which is translated into proteins.