So what's the right word?
Project - Sam gives good reason why this word works in the beginning, so long as you're not looking for external validation. At some point though, your project becomes something more.
Company - the people working on the project, and all the processes and resources that go into supporting those people.
Business - the transactional model that enables the company to make money and keep doing what they're doing.
Mission - for some people, the business is the mission. But for most success stories, even in the beginning when it was just a project, there was an underlying mission, a purpose and a plan. Unfortunately, the word feels mushy. I'd hate to hear people going around saying they're "working on a mission."
Ultimately, the word I use will likely depend on the context.