Captains are like the captains of ships, and especially like the romanticised captains of spaceships in SF series. They decide where the thing and everyone in it will go, fight nonsense and complacency, and lead everyone else in overcoming or escaping their enemies. Captains breathe soul into the ship.
For most people, captains are the role that best fits the concept of a leader. However, the main function of captains is not to rally people behind them or to motivate them. It is to take decisions based on some vision of the future, good or bad, and implement them into the present where that vision is not at all apparent. To put it another way, captains make the important decisions before conditions change to make them plainly obvious to everyone.
You need a captain to get you out of a forest, or to find a new continent. Their main job is to show that there is an exit, or a new world, convince at least some people to come along, and then get there against the inevitable doubt and adversity. Captains may lie about actual conditions or be very frank, but in either case they rate people according to their motivation and contribution to the goal.
In business, captains tend to be the founders of whole companies, or of maverick business units within companies led by claimers. At least some businesses led by captains do extraordinarily well (Google, Apple, Microsoft). The role of captain is generally not advertised or appointed. You get it by creating your ship.
Common downfalls of captains are to become emperors or prime ministers. Emperors manage to engender a very strong belief in the edifice they have created, while losing their grounding on reality until they becomes isolated or fall down in flames. Prime ministers over-emphasise attracting followers, until they have a great number of people behind them but lead them nowhere in particular.
Optimisers are the people who run things. Their role is to act as mentors, coaches, counsels, and generally to bring knowledge about the domain as well as personal care into an endeavour. They maximise the the long-term welfare of whatever they're in charge of, which means looking after people and resources while gradually achieving higher and higher goals.
The day-to-day activity of optimisers is to look at the working whole from their particular point of view and try to make it better. Crucially, they cannot cut away their charge from the whole, or tear it down and restart it in a radical way. They have to look at all aspects of the working web, people, technology, interactions, and make them better by introducing an improvement, or fixing something that is bad. They go about this this in a very hands on way, by showing or doing.
An optimiser is the right sort of person to lead you up a mountain. You don't need a captain for that because mountains are generally easy to see, and everyone can start walking along the side and make progress. However, an optimiser chooses the best path, makes sure everyone is well equipped and no-one is falling behind, gets people to use ropes in dangerous places, and ensures someone has brought the food.
It's natural for engineers in particular to grow into optimisers, and they ought to be the most numerous type of manager in a productive business. They are the force that preserves the health and productivity of an organisation against decay or neglect. It's the role of optimisers to make everyone in the organisation happy, but it's not their role to make them want to be there in the first place.
The main way in which optimisers fail is by stagnating or growing out of the role. From the the individual, stagnation manifests as lack of interest or application, and a tendency to adopt more and more procedural, institutionalised, and generally defensive approaches. The organisation also has a tendency to grind good optimisers out of their role, usually turning them into bad claimers.
Claimers are the big bad bosses. Their role is to enforce the control of some external entity onto an organisation, and get the organisation to do what the external entity wants, rather than what the organisation itself wants, to the best of their abilities. Claimers represent owners or overseers, and they do so imperfectly but firmly.
The work of a claimer is to set some active goals or some restrictions, tell people in a more or less forceful manner to meet them, and then demand proof that the goals have been met. The claimer does not, in general, observe the workings to ensure first hand that goals are met, because that's the optimiser's job. The claimer instead summons those who run things to show their results, and on the basis of that hands out rewards or penalties. This form of control is acknowledged by all as imperfect, but necessary to align the organisation with its owners.
If a claimer is needed, in the great scheme of things, it is to lead people through a prairie. Their role is to force people to move forward at good pace, not play with the animals or lie under trees, and most importantly to make their way to the mines up the hills rather than roll in the opposite direction to the beach. This function may be good or bad depending on your point of view.
In modern businesses, claimers are obviously there to represent the interests of the shareholders. They may fit more or less harmoniously, but they are fundamentally antagonistic to the whole of the rest of the organisation, which by itself wants to pursue other goals such as the welfare of employees or random idealistic targets. Claimers are always present at lower levels, such as HR accounting and the like. At a high level, it it's a defining characteristic of a business whether it is run by a claimer or a captain.
The chief failure regarding claimers is having too few or too many. Having too few is rare, and results in squandering or resources in the form of failed dot-coms. Having too many is endemic in business, the more so the larger and more traditional the business. This chokes the business, not because the claimers themselves are unproductive but because the blunt and antagonistic form of control that they impose destroys productivity.