How does the DNA inherited by an organism determine the specific traits displayed by that organism? That is, how does the specific order of nucleotides in the DNA inherited by a critter determine its phenotypic characteristics such as eye and hair color, the kind of digestive enzymes produced, whether wings or arms will be produced, etc.?

When it comes to how DNA determines the traits displayed by an organism, there is a very specific process involved in doing so. It is known by the term "gene expression." There is a whole process involved, as stated above; this process mainly consists of transcription and translation. Transcription involves the interpretation of a DNA into an mRNA strand with the help of numerous enzymes -- RNA pol II, helicase, topoisomerase, primase, etc. There are also post-transcriptional steps involved -- but only in eukaryotes (not prokaryotes!).

Translation is the interpretation of an mRNA strand by organelles called ribosomes. Ribosomes are made up of two main subunits -- a large and small. These subunits are composed of rRNA and protein. When an mRNA strand is translated, a polypeptide is formed. A polypeptide is basically a chain of linked amino acids. It can be a stand-alone protein or part of a larger protein. Main idea is, however, that a protein is formed, and collective groups of protein establish phenotypic characteristics.

There is also gene regulation involved. For instance, bacteria utilize sequences on their DNA strands known as operators, as well as regulatory proteins for gene regulation (activators and repressors). When a repressor attaches to an operator, the operon (the set of genes + promoter + operator) is switched off, not activating genes. This is a whole entire topic on its own.

Hope you found this helpful :)