What is the relationship between the sequence of bases in DNA and the sequence of amino acids in proteins?

DNA is a double-stranded macromolecule that serves as an instruction set for making all of the proteins that can be made by a cell. The instructions are determined by the sequence of the nitrogenous bases (adenine-A, thymine-T, cytosine-C, and guanine-G). The DNA has two strands:  a sense strand and a template strand. The sense strand is typically the one drawn on top, going 5' to 3'. The template strand is typically drawn on the bottom, in the 3' to 5' direction. It serves as the template for RNA polymerase during the process of transcription.

Transcription occurs in the nucleus of a eukaryotic cell and is the copying of the DNA into an mRNA message that can go to the ribosome to make a protein. In eukaryotes, the DNA cannot leave the nucleus so a messenger is needed. After the DNA is unwound and the two strands are pulled apart, the RNA polymerase synthesizes an mRNA strand by putting together a sequence of bases that are complimentary to those of the template DNA strand:

Sense:  ATG  CAT  GAT  TAT  CCC  ACT

Temp.:  TAC  GTA  CTA  ATA  GGG  TGA

-You can see that in DNA A pairs with T and C pairs with G to make the two strands

The template strand is then used as a template to make mRNA. RNA does not contain thymine and in its place there are uracils. Cs still pair with Gs. Notice that mRNA is single-stranded while DNA is double stranded.

Template DNA: TAC  GTA  CTA  ATA  GGG  TGA 

mRNA:            AUG  CAU GAU  UAU CCC  ACU  

After processing and the addition of a 5' cap and a poly-A tail, the mRNA can leave the nucleus and go to a ribosome where a protein is made via translation.

The ribosome reads the mRNA 3 bases at a time. These 3-base segments are known as codons. Based on the codon, the ribosome will bring over a particular tRNA carrying a specific amino acid. The anti-codon on a tRNA will be complimentary to the codon sequence of the mRNA and this ensures that the correct tRNA is used. Each mRNA codon specifies 1 amino acid or stop. You can read the mRNA sequence 1 codon at a time to figure out the amino acid sequence of your polypeptide chain.

mRNA:  AUG  CAU  GAU  UAU  CCC  ACU

using a codon table (see link below):  

AUG:  methionine

CAU:  histadine

GAU:  aspartic acid

UAU:  tyrosine

CCC:  proline

ACU:  threonine

The ribosome forms peptide bonds between the amino acids in the chain.