A division of a business entity is a portion of that business that operates under a different name. It is the equivalent of a corporation or limited liability company obtaining a fictitious name or "doing business as" certificate and operating a business under that fictitious name. A division is different from a subsidiary, in that a subsidiary is a separate legal entity owned by the primary business. A division is a part of the primary business, is not separate from that business, and the primary business is legally responsible for all of the obligations and debts of the division.
Generally, only an "entity", e.g. a corporation, limited liability company, etc. would have a "division"; an individual operating in this manner would simply be "operating under a fictitious name".
An example of this would be to look at Hewlett Packard (HP), the computer and printer company. HP has several divisions, with the printer division, that makes laser and inkjet printers, being the largest and most profitable division. The divisions of HP, like the Printing & Multifunction division, the Handheld Devices (includes the old calculator) division, the Servers division (mini and mainframe computers), etc., all use the HP brand name. But, Compaq (a part of HP since 2000), operates as a subsidiary, using the Compaq brand name.
Mack Trucks continues to run itself separately, but is a wholly owned subsidiary of AB Volvo. Volvo Trucks also sells trucks under the Volvo name. Within Mack, there is a division named Mack de Venezuela C.A. that does final assembly and sells trucks in South America.
Another more obvious example is that Google Video is a division of Google, and is part of the same corporate entity. But the YouTube video service is a subsidiary of Google because it remains operated as YouTube, Inc., a separate corporation even though it is owned by Google.