One of the key concepts to understanding how a single celled organism can develop into a differentiated multicellular organism is to understand the central dogma. The central dogma is a term that describes how cells use DNA to produce mRNA to produce proteins. To produce a particular protein, the gene must be turned on and transcribed into mRNA, which then must be translated into protein. It is also important to distinguish between genotype and phenotype. The genotype of an organism is the genetic information (genes) it acquired from its parents. The phenotype describes how the organism appears physically, including its internal structure, and the biochemistry of its cell(s). The dynamic relationship between the genotype and phenotype is what attributes to the development of specialized and different cells. For example, consider why one of your liver cell looks different, and functions different from one of your heart cells. If the DNA in these cells is the same, how is it possible that they look and function differently?
The answer to this question is regulation. Although this liver and heart cell contain the same exact DNA sequence, not all of the genes in the DNA are turned on and producing the same exact proteins. What a cell can do is largely determined by what proteins it contains. In order to "turn on" a gene, a combination of steps must happen, which may include the binding of a specialized gene-regulatory protein to the control regions of the DNA. Both transcription and translation are regulated by multiple other mechanisms as well. For example, some mRNA may get degraded before it has a chance to be translated (preventing the protein from being produced.) Another example is called RNA Processing. After the gene is transcribed to RNA, the RNA may be cut and spliced in different ways to give rise to many different mRNAs (and proteins, all from the same gene). Alternatively, the proteins produced can be modified after translation as well (called post-translational modifications) before they become active proteins. One last mechanism is a category of genes that produces microRNAs. These are little RNA fragments that prevent the translation of specific mRNAs.
Together, all of these mechanisms mean that one gene can produce a multitude of functionally different proteins. So although the liver cell and the heart cell have the same DNA sequence, this does not necessarily mean they will be producing the exact same proteins. By controlling which proteins are made in the cell, the genes (DNA) can control the properties and behaviors of the cells, which determine the course of development. Genes control the development of the organism by determining where and when proteins are made (there are many genes involved in this process). The interactions between proteins and genes and between proteins and proteins also contribute to the properties of the cell.
One of these properties is cell-cell interactions, which is the ability of a cell to communicate with, and respond to, the cells around it. Their response to signals for cell movement, or change in cell shape brings about morphological (physical) changes in the developing organism. The cell-cell interactions are what determine how the organism develops (where the heart goes in relationship to the liver, arm, etc.) so therefore no developmental processes can be due to the function of a single gene or protein. Aside from just a single cell influencing the actions of another single cell, this can also happen in groups. One group of cells can influence the development of an adjacent group of cells. This signal can be sent through many cells, or be localized to a few cells.
Overall, the differences between cells must be generated by differences in gene activity which would lead to the production of different proteins. This leads to differences in cell properties and functions. Different proteins located on the outside of the cell, or being released from the cell provide a mechanism for cell-cell interactions/ or communication between cells. Responses to these signals can bring about physical changes in the developing organism, thus producing two cells with different physical properties and functions, but that still contain the same exact DNA sequence.