Something important to elementary and for elementary OS is our branding. For a quick reference:
"elementary" is the name of the company. It is never capitalized. Refer to "elementary" when you're talking about the company itself or the developers. Think of "elementary" as the group of people.
"elementary OS" is the name of an elementary product. "OS" should be capitalized as it stands for "operating system". The company name is still not capitalized. There is a space between the company name and the product name. When referring to the product, always use "elementary OS" and not "elementary".
"eos" is the name of a line of beauty products or Canon cameras. We don't consider this an acceptable way to refer to elementary OS. This is especially important in not encouraging 3rd parties to make brands such as "eRadio" or "eCleaner".