A small business is a business that is small in terms of employees, revenues, or geographic range. A small business is also usually privately owned. What exactly constitutes a small business is open to interpretation, but it would usually be described as a business where the owner or manager (who would often be the same person) would be personally familiar with all his personnel, equipment and in some cases, customers. Most small businesses are in either the service or retail sectors, or else in some type of craft or light manufacturing.
Having described some of the denotations of what a small business is, I should say that they are not really important, because the term "small business" is mostly used politically for its connotations. If you read almost any press statement from any mainstream politician, "helping small businesses" will be one of the positions that always comes up. It is similar to "investing in education", or "giving a livable future to our children" or the infamous "job creation". In fact, as a politician, if you want to stop discourse, just state that you wish to "invest in education so that small businesses can create jobs", and no one can refute you, but that is getting a bit off-topic.
At least in the United States, small businesses are popular, at least the image of them, for several interrelated reasons. The first is that most Americans, at least as a ground state of belief, do not believe in heavy forms of socialism or government planning in the economy. Most Americans believe nominally in capitalism and free markets, (even though these are mutually contradictory terms). On the other hand, culturally, most Americans also don't like big business. To most Americans, big businesses has images of greed, heartlessness, bureaucracy and conformity. Most Americans have, at least in passing, worked for a big business and know what it is like. And if not, they have at least interacted with the bureaucracy of a big business, and found it not to their liking. And thus appears the hero of the story, the small business. A small business is the creation of free enterprise and hard work, and is sensible with its money without being greedy. The owner of a small business probably lives in the same town as you, and might do some volunteer or community work. They may help people in the community by offering employment, or donating services in kind to a school or neighborhood group. And while most Americans can't hope to someday be a plutocrat, and would probably disclaim wanting to be, most Americans can hope that someday they might be able to open a small retail store and earn enough money to own a vacation home. So when the government talks about helping "Small businesses", it encourages people, because that business could conceivably belong to them, or at least to someone they know.
There are a few problems with "Small businesses". The first is, even though there is a lot of small businesses, and many people are employed by them, they can't do some very core tasks for the economy. A small business can run a bakery and bake bread, but a small business can't buy a freighter and ship wheat across the Pacific Ocean. A small business might be able to build you a custom computer, but a small business can't design and manufacture a CPU---you need Intel for that. Small businesses tend to be middlemen, rather than primary producers. For primary producers, the economy of scale seems to suggest multinational corporations are better. But most politicians, even conservative ones, won't brag about giving tax breaks to Intel, even if it benefits people in the long run. In other words, the cultural connotations of "Small Business" tend to drown out the economic ones.
The Small Business is used with some dishonesty by politicians on each side of the aisle. For liberal politicians, pro-business legislation can be passed without fear of populist backlash by reassuring the base that the businesses being helped will be on a scale that is not too far from normal, working class people. For conservatives, "Small Business" is a magical point where cultural values and economic beliefs meet. It can be used to avoid the fact that market forces often encourage larger businesses to succeed over smaller businesses, just like the fact that market forces often eliminate jobs instead of create them is often ignored. In either case the comforting images of small businesses are used to gloss over real conflicts and uncertainties in people's economic futures.