It's not clear to me if you want a definition of an emerging sector or if you want an example of an emerging sector. I'll give both.
Emerging sectors in an economy are sectors (types of markets) that seem to be getting more important in that economy. These markets seem to have the potential to grow.
As for examples, this varies from country to country. Worldwide, the most likely emerging sector appears to be "green power." Companies that have good products in this sector should be able to experience rapid growth in the near future.
Since you posted this in the economy sector, I will assume that you mean areas where there might be some growth in the future. Well, first let me start with emerging places. I think Asia will be more of powerhouse in the future. We already see this with China, but other countries in South East Asia will get stronger in my opinion.
In relation to this, I think that commodaties will be increasingly important, oil and metals, in particular. As these nations build their infrastructure, they will need these things more. Also, when you consider oil, there is only a limited supply. So, prices should go up. Another potential for growth may be agriculture.
Emerging sector in business refers to an industry within an economy or a country that is in its initial stage of development, and is expected to grow rapidly in coming few years. The word sector may also be used for a group of industries. For example, economists generally divide all the industries in an economy in broad groups such as agricultural, manufacturing, and service sectors.
It is observe that every industry within an economy passes through a set pattern of growth and decline. When the industry is just founded and is in its infancy it experiences a very slow rate of growth, and its total size is also very small. As the industry develops the rate of growth slowly increases and over a period becomes very high. When the rate of growth is highest the size of industry is also substantial. After this stage the growth rate begins to taper down. At this stage the industry continues to grow in size, but at ever declining rate, till it reaches a maximum level when the industry is stagnant. Following this the growth rate becomes negative and the industry becomes passes in to stage of decline.
An industry will qualify as an emerging sector, when it is still small, but its growth begins to accelerate. The size of the industry is not yet big, but is expected to become so in near future. An industry qualifies as emerging sector, not only on the basis of its existing stage, but also on expectation of future growth. Thus it is not possible to identify emerging sectors of an economy with certainty. There are bound to be some disagreement among experts on the list of emerging sectors of an economy. I believe that some of the emerging sectors in Indian economy are organized retail, health care, insurance, and knowledge process outsourcing.