Business analysis is a process for uncovering business goals, shaping them in clear and understandable terms, and identifying and evaluating a range of solutions. The business analyst's job starts with activities aimed at analyzing and understanding the goals the business wants to achieve.
Of course, this will probably require interaction with many representatives of the business, to understand how the business works, and how it delivers its products and services. With a firm grip on how the business functions, a business analyst will be able to start identifying and planning any changes the organization must make in order to meet its goals. This is often the most rewarding part of a business analysts' work—helping to define a range of possible solutions, evaluating the effectiveness of any proposed strategies or tactics, and helping the organization understand which one(s) will best help it achieve its goals.
So the business analyst role isn't fuzzy or mystical at all. In fact, anyone engaged in the activities outlined above is conducting "business analysis." But because doing the best possible job of business analysis requires special analytical and communication skills, it has emerged as a full-time profession. Although most companies do at least some form of business analysis daily, the professional business analyst's role is to conduct business analysis activities methodically and with deliberate care.
Business analysts tend to serve as liaisons between the various parts of an organization. A common example involves the analyst interacting with several operational business teams to understand their business goals and define specific steps they can take to achieve them. The business analyst then communicates the needs to a development team tasked with building the solution. Finally, the business analyst evaluates the degree to which the solution has met the needs. Thus, this role is often considered a "translator," because they translate the needs of the business into specific deliverables that the various teams can interpret and act on.
To sum up: serving as a business analyst means first understanding the business and eliciting, uncovering, and defining its organizational goals; then identifying the specific tactical steps the business must take to meet these goals; and finally, transferring the gathered knowledge about the business needs to those in a position to craft a solution. When the project is finished, the business analyst evaluates the extent to which the solution has met its target business goals.