The word "organic" in "organic t-shirt" refers to the type of cotton that is used in the shirt's creation. In order for a t-shirt (or any piece of cloth or clothing, for that matter) to be considered "organic" it must be created from natural materials (cotton, wool, hemp...whatever) that have not been treated with a chemical fertilizer or pesticide. In addition, for a piece of clothing to be truly "organic," it must be made from plants that are not genetically modified.
Organic t-shirts are created by the same processes as regular t-shirts, the term simply refers to the material used in the shirt's creation.
Organic t-shirts are cotton t-shirts made form organic cotton. To qualify as organic cotton the cotton must one of the natural varieties, as opposed to genetically modified ones. Also organic cotton is grown without using any synthetic fertilizers pesticides.
As far as physical characteristics of organic t-shirts are concerned there ins no difference between these and other cotton t-shirts. Also the process of making organic t-shirt goes, it is exactly same as that used for making cotton t-shirts in general.
Organic t-shirts are made and marketed to appeal to people that are opposed to use of synthetic fertilizers and pesticides in farming and to growing of genetically modified products. In general these are also the people who oppose use of technologies and manufacturing processes harmful to the environment. Therefore, companies may also frequently claim to use green technologies. But there are so many different processes are involved in production of a t-shirt that, I am not sure if it is possible to be 100 percent green. For example, I am not sure if there is a way to bleach cotton cloth without using chemicals, or that buttons used in organic t-shirts are same as the buttons of synthetic material used in common t-shirts.