When an organization has multiple jobs for which you could do a great job, how do you choose what to apply for?

I think it can be attractive if done in balance to communicate to a firm that you've admired them, and have skills and experience that could fit into a number of roles with them. While it doesn't happen to every employers, iconic employers (think Apple, and formerly firms like Levis or Gap) are used to it. Two small cautions: 1) Taken too far this keen interest in a firm can be perceived as "stalking" and 2) you want to be able to communicate both interest in the role in the firm so that you avoid a potential perception that you're not as qualified as someone who leads with the job, not the firm. Companies should feel honored that people want to work with them. Some do, and some turn that interest to a sort of warped elitism.

When applying for a job at an organization where there are several positions in a single department you should try to connect with the manager to discuss the different needs of the team to determine which role you would be a better fit in - before sending in an application. If you are interested in positions in different departments it is a more difficult task. An individual hiring manager for a single department generally wants to hire a candidate who wants to work in the managers department, not 'any' department in the organization. That is the potential danger of targeting too many departments - none feel you actually want their department. Whenever possible target one department, one team and one hiring manager at the outset. Connect with and network and/or informational interview that department manager. Express your interest in the entire organization, but your research has taken you to that department first as you saw it as the best fit. During the course of that process if you find that there are no relevant openings you can ask the manager to suggest other departments and managers and ask for a reference to those managers.  If you can not clarify what department or team really is the best fit then you may have to throw a dart at the job posting on the wall and allow serendipity to be your guide as to where to begin - or you can see what network connections you have and take the best lead as your first contact. However you approach the situation remember most managers want to be your first choice, not your 'as good as any' choice.