We use the term "project manager" a lot. Of course, this role simply refers to the person that manages a project. You may not have the title of a formal project manager in your organization, but the project manager is the role of the person managing the project.
That part seems pretty cut and dry. However, the way that organizations utilize project managers is anything but consistent. I bet if I talked to ten readers, you would all have different ways that you utilize project managers. Because of this variability, the role is usually categorized in one of three ways, depending on the relationship between the project managers and the functional managers in your organization.
"Weak" project managers. In this model, the functional managers control most aspects of projects within their organization. The functional managers control the resources, manage the budgets and schedules, and are responsible for overall project success. The project may also have a project manager, but the person really has little authority. Perhaps their role is to update the schedule and collect status updates, but the functional manager is making all the important decisions.
"Strong" project managers. This is the other extreme. In this model the project managers are 100% responsible for project success. They manage schedule, budget, quality, scope, etc. They may have their own staff or they may have to get staff from functional managers. However, once a staff member is assigned to a project it is clear that the project manager maintains overall control of the resource until the work of the project ends.
"Balanced" project managers. In this model, the relationship between project manager and functional manager is pretty much equal. In other words, many decisions are made jointly. For example, the project manager may get resources from functional managers, but it involves more give-and-take between managers that are peers. Similarly, if a team member needs to be trained, there is a negotiation to see whether it makes sense for the training to be paid for by the functional manager or the project manager.
The reason this is interesting is because in many organizations, the job description of the project manager role doesn't line up with the actual responsibilities. For instance, your organization might state that project managers are responsible for the success of projects, and yet they don't have responsibility for budgets and they don't have much control over staff. In that case, it is much harder (but not impossible) to be successful.
It is very possible that functional managers will make most project decisions. There is nothing wrong with that in itself. However, in that type of model (weak project managers) you should not pretend that a project manager is responsible for project success. If you do, then you will have a disconnect between what you say and what you really do. This will lead to "frustration culture" as described in an earlier Plain Talk email.
For what it is worth, generally the more mature an organization is, the stronger the project manager role. Usually the functional managers are in control of projects because an organization does not understand the role of project manager, they don't respect the role, or they simply are unwilling to turn over authority and responsibility to the project manager role. As an organization develops a more mature project environment, the project managers generally become more important and have more responsibility for the success of projects.