The idea of stem cell "self-renewal" is actually confusing, and somewhat misleading - it is not that stem cells renew themselves, rather that they do not differentiate. 

Cells differ in the sets of genes that they "express", that is, which genes are transcribed into RNA and which gene products (often proteins) are active. 

The set of genes expressed by a cell is dependent upon its history (which genes were expressed previously, the state of its chromatin, etc), its current environment, and the signals that it receives from that environment.    

The major difference between stem cells and other cell types is that, because of the genes they express, stem cells can, in theory at least, divide an unlimited number of times. 

Non-stem cells other have a "limited proliferative life span", if they can divide at all.   

Why? Because stem cells are specialized to divide but not differentiate, whereas other cell types are committed to form specific types of cells.  

When a stem cell divides, one daughter remains in the original environment (or niche), while the other moves into a new environment.  As a cell moves into a new environment, the pattern of genes that it expresses changes - it begins to differentiate. 

The changes in gene expression can also limit the cell's ability to divide.  For example many differentiated cell types, such as neurons, cardiac and skeletal muscle cells, most skin cells, and blood cells, are "terminally differentiated" and cannot divide ever again.

There are two generic types of stem cells in mammals, embryonic and adult. 

Embryonic stem cells, which are derived artificially from the inner cell mass of the early embryo, can produce all somatic cell types. 

Embryonic stem cells are also involved in the generation of new organisms. 

A number of different types of stem cells are present within the adult organism and produce only a limited set of cell types.  

For example, skin stem cells are located a region of the hair follicle,  These cell divide and one cell remains a stem cell and the other migrates away. 

They cells that migrate away go on to divide and differentiate, forming the other cells of the skin.  This process regenerates the skin. 

In an analogous manner, there appear to be stem cell populations that maintain other organ systems, such as skeletal and cardiac muscle, the brain, and the hematological system. 

So, an obvious question, which I think is close to the one you are asking, is "why aren't all cells stem cells?"

There answer is thought to do with the need to tightly regulate cell division.

With each cell division (and replication of the genetic material), cells accumulate mutations; in some cases, these mutations can lead to unregulated or inappropriate cell growth. 

Unregulated cell division is the basis for cancer, and can lead to the death of the organism. 

It is thought that restricting the proliferative life span of most cells in the body is a anti-cancer strategy; unfortunately cancer cells can find ways to reverse this restriction on cell division. 

For example, stem cells have active telomerase while most non-stem cells have turned off the genes encoding telomerase.  

Telomerase activity is involved in protecting the ends of the chromosome from "molecular erosion" during cell division – when this occurs, it leads to cellular senescence.  In many cancer cells, the genes for telomerase have been turned back on.