The generic word is "logo" for a symbol that represents a company, service, product, etc.
Logo is actually short for "logotype" or "logogram" which literally mean struck or written word. Of course a logo doesn't necessarily consist of, contain or stand for a word at all. "Logo" "Brandmark" "Trademark" "Symbol" "Mark" "Service Mark" are more or less synonymous.
There are several different types of logos and it helps in thinking about or designing logos to know from the beginning what kind of logo you are looking for. There are different terms used but I like this classification:
1. Words: The company or product name, presented in a certain color, font, etc. (Coca-Cola, Disney, Tide)
2. Emblem: Company or product name as part of a unique graphic representation (Starbucks, Harley-Davidson).
3. Pictorial: A stylized image that becomes recognizable as a representation of the company, product or service (NBA, Apple)
4. Abstract: At most loosely based on anything real (Nike, Mercedes-Benz)
5. Letters: One or more letters represented in a unique graphical form to stand for the company, product or service (Unilever, GM)
6. Combinations of one or more of the above. The AT&T symbol is pictorial, but is usually presented with the company name, similarly for the NBC peacock.