Let's see a simple explanation, eh?
First consider that at conception a human is a single cell, and from this single cell, hundreds of different cell types arise. Basically, this means that that one cell has the potential to become all the different types of cells that will eventually form a living human.
As the organism ages, the cells split and "differentiate" into cells that are more specific (for instance myoblasts which give rise to many different types of muscle cells, or cells of the ectoderm which give rise to skin and neurons).
The more differentiated the cells become, the more limited are the cell types they can become.
So in the end, fetal stem cells are the best to harvest because they have the potential to become any human cell type if they are exposed to the right combination of protein signals. Adult stem cells are more limited because they have further differentiated and can only produce related cell types (for instance, the stem cells found in bone marrow cannot be used to produce neurons).
Stem cell research is basically taking various types of stem cells and exposing them to proteins or chemicals in order to get them to produce specific types of tissue to use for therapy or repair of people. Also attempts are made to use stem cells directly by placing them at a site of disease or injury in the hopes that they will differentiate into the cell type that has been damaged (like putting stem cells in a damaged spinal cord hoping to repair it).
An undesirable finding is that since scientists don't fully understand the signals that determine exactly what stem cells differentiate into, sometimes therapies end up producing cancerous tumors instead of the desired tissue.
Hope this is what you're looking for.