'Small business' is a very broad term for a huge range of different types of business career. It can mean a retail shop, a trades business, a creative cottage industry business, a service business or almost anything else. The actual definition of small business usually means in practice a sole proprietor or partnership in a purely local business.
Most people enter small business by starting a business venture on their own. They may either create a business, or buy a business, and try and develop it. Most small business people do have some level of business training, and they need it. Although professionals like accountants can help with some things, small business owners really do need to know how to conduct business properly. Small business involves a very wide range of business practices, which are both necessary and required by law.
Career advancement in small business equates to the development of the business. Many small businesses start out as very basic operations like retail, or single stream service or professional businesses. Successful small businesses evolve over time and generally take on many more roles expanding their market base.
As the business evolves, so, naturally, do the commercial opportunities and the requirement for the people running the business to upgrade their skills. Many of the world's biggest corporations, like McDonald's, actually started out as small businesses. McDonald's used to be one single hamburger shop, and within a generation and became the biggest, best known corporation in the world.
Not every small business turns into McDonald's. There are many risks, and a large percentage of small businesses fail within the first three years of operation. It is no coincidence that the most successful small businesses are those operated by people having extensive training as well as strong financial and business management skills.
If you're looking for a career which can provide huge challenges as well as huge rewards, you are very strongly advised to research the career possibilities of small businesses. A lot of self-made millionaires started out operating small businesses, and succeeded on the basis of their business skills. If you know how to manage risk, this may be the career you want.