The "corp" part of incorporated is from latin for "body". When you make a corporation, you are creating a new legal entity--legally, you are giving your company a body separate from yours. With an unincorporated business, the business is just how you make money, but the company doesn't actually exist--when people deal with the company, they are really dealing with you. With a corporation, they are dealing with a company, maybe a company that you own and are the president of, but still a separate company and not you personally.
It makes a difference mostly in accounting and liability. Consult a lawyer if you need to know for practical reasons or are trying to set one up--there are a lot of details you can't get here that need to be considered. The big difference is that with a corporation, your company will secure loans, sign contracts, pay taxes, do work and be responsible for it, and all the other things business do, rather than you. For instance, if your corporate business goes bankrupt or gets sued for more than it is worth, you would just lose the business. If your unincorporated business fails and owes money, you would have to go bankrupt yourself; if the unincorporated business gets sued, then you would actually be sued, because the business doesn't exist to the court--and if the company's money can't afford the damages, your house (in some places, not others) and other private assets could be used to pay them. The minus to having a corporation is that the corporation pays taxes, pays you a salary and bonues or dividends, and then you pay income taxes on the money paid to you by your company that your corporation has already paid income taxes for.