Between different consulting firms and practices, the job responsibilities and experience levels of engagement managers vary widely, but there is one characteristic of the position that I like because it captures the essence of management consulting better than do characteristics of other typical consulting positions (e.g., principal, partner, director, associate, manager). By understanding the central function of "engagement management", one can better understand the essence of management consulting.
Engagement managers own the problem statement from the perspective of the customer, and thus, have the responsibilities to ensure that consulting team both structures the problem solving methodology correctly and executes on the problem solving methodology.
Thus as an example, a problem statement may be to help figure out whether a client should enter a wireless business, identify under what circumstances it makes sense, define the strategy and plan for how it should be done, and get cross-functional buy-in from the management team and Board of Directors. In this case, the engagement manager may need to work with the consulting team to synthesize primary and secondary marketing research from end users and distributors, construct financial analyses, develop technology scenarios and architectures, conduct client workshops on various subjects to gain insights and share best practice perspectives, perform gap analyses between present methods of operations and desired future states, or perform competitive analyses and forecasting.
Another problem statement might be, "figure out the root cause of declining customer satisfaction and fix it because my internal management team is giving me mixed messages". Yet another one might be, "how do I transform my business from doing lots of low margin X to doing more high margin Y". (As a quick digression, one should note that setting up the problem statement properly is very key to selling consulting engagements and solving them – do not take it for granted that the problem statement is articulated properly).
In essence, the role of the engagement manager is to help the client to solving their problems by synthesizing the work of smart people and subject matter experts in different functions and areas from throughout the firm.
In closing, I find that people frequently confuse the role of project managers with that of engagement managers. There are definitely some overlapping functions, but the essence of project management is more to ensure that things are accomplished on time, on budget, and according to customer specifications. On the other hand, the role of the engagement manager is to own the client problem statement as if it were their own. At risk of sounding like I am diminishing the importance of the function, project management becomes more of an execution detail in the greater scheme of things.