Proteins are synthesised through the concept of central dogma. The central dogma states that first the DNA is converted into mRNA by transcription, and then by translation the mRNA is converted into proteins. There are segments of DNA that are called genes which code for a particular protein.
Transcription: In transcription that occurs in the nucleus, the DNA is transcribed into mRNA. DNA is a double stranded molecule with A, T, G and C nitrogenous bases while mRNA is a single stranded molecule with A, U, G and C bases. The DNA molecule unwinds with the help of an enzyme that breaks down the hydrogen bonds between the complementary bases (double bonds between A and T and triple bonds between G and C). Only that portion of DNA unwinds for which the gene is supposed to be transcribed. Then, on a template strand the nucleotides are exposed and free nucleotides then form complementary base pairings with them. Then the RNA polymerase enzyme forms phosphodiester bonds between the nucleotides. Then, the mRNA breaks off and leaves the nucleus into the cytoplasm via the nuclear pore.
Translation: The mRNA formed goes and binds to the ribosomes in the cytoplasm. The mRNA has codons (like a start codon AUG, a set of three bases) which code for a specific amino acid like AUG codes for methionine. Also, tRNA is present in the cytoplasm. At one end, there is anticodon that is complementary to the codon of the mRNA, and at the other side there is an amino acid. When the codon and anticodon bind, the tRNA leaves its amino acid at the ribosome, and the peptide bonds form between the amino acids. The polypeptide chain continues till the stop codon (UAA, UGA and UAG and they code for no amino acid) comes and the polypeptide chain is released.