Gestalt Psychologists were interested in complex phenomena: How people perceive scenes and spaces; how they solve complex problems; how they relate different components of their experience with one another whatever the domain of the experience. Gestalt Psychology is usually contrasted with Associationism, Wundtian or Titchenerian Structuralism, and Behaviorism. Their focus is on forms, structures, configurations, or wholes rather than elements. According to Gestalt Psychology one must interpret "elements" as parts of larger structures rather than as independent atoms which combine to form larger units. They differed in their methods from their competitors. They substituted phenomenal experience in place of Wundt’s and Titchener’s analytic introspection and they studied complex responses to complex stimuli rather than unidimensional responses to simple stimuli. Whereas other systems thought of learning as gaining new connections between elements through association, they thought that the best learning (problem solving) was due to insight, or restructuring the components of a whole so that they could be seen or understood in a new way. This restructuring was often thought to happen holistically and suddenly. They were particularly interested in the fact that the same stimuli may be perceived in different ways. One consequent of this interest is that they studied illusions. Click for some illusions and more illusions

Gestalt psychologists position had many precursors. The more influential ones included:
Ernst Mach, who claimed that there were sensations of space-form and time-form.
J. S. Mill, who argued that one could not find the causal elements of a phenomenal experience directly in the experience.
Christian Von Ehrenfels (1859-1932), a student of Brentano, who invented the concept of form quality  (Gestaltqualitat), which was a larger unit, such as a square or a melody, that did not inhere directly to the elements from which it was derived.
Albert Einstein, who proposed that the world could be understood in terms of field dynamics rather than in terms of independent atoms.

Gestalt Psychology has been an important influence to the development of much modern psychology. Some of their claims, particularly their general disregard of learning and experience are not generally accepted today, and their neuropsychological claims are quite controversial, but much of their work has become foundational for modern psychology.

Max Wertheimer (1880-1943)

Max Wertheimer was the primary founder of Gestalt Psychology.
He studied with Stumpf and Külpe
Began in 1912 with a description of apparent motion or phi phenomenon. The essence of the claim implicit in phi is that phenomenal experience cannot be broken into sensory components. The stimulus for phi movement does not exist in any direct sense in the physical world.
From this beginning many aspects of perception were shown to be phenomenally real, but not physically present.
Wertheimer also did important research in thinking and problem solving. His posthumously published Productive Thinking (1945) is still in press. Thinking is best done when the parts and wholes are seen in a coherent structure. He identified most teaching of thinking as "stupid" because it did not adhere to the intrinsic structure of the problems that were to be solved, but rather was taught as memorizing formulae that when applied correctly gave the right answer. He believed that if a student learned the intrinsic relations between parts and wholes, they would remember better and be able to generalize better.
Wertheimer and all of the early Gestalt psychologists argued that psychological phenomena of perception and reasoning are based much more on the structure of the phenomena than on specific experience. What was perceived was directly a function of the structural relations in the stimulus. And in order to fully understand one must see the relationships that inhere in the phenomena.
Wertheimer's Gestalt Theory and Laws of organization (1923)
Wertheimer, Koffka, and Köhler were the primary originators and developers  of Gestalt Psychology. These three scholars seem to be in agreement with one another. They worked independently but pushed very similar agendas.
 

Student of Stumpf
First introduced Gestalt psychology to America. He used perception as his examples so many people think of Gestalt Psychology as being simply a theory of perception.  Perception: An introduction to Gestalt Theory . It contains a glossary.

He wrote Principles of Gestalt Psychology in 1935, an excellent systematic thesis on Gestalt theory, applying it to child development, social psychology, thinking, learning and perception.

Student of Stumpf
The most popular of the principle Gestalt psychologists, probably because of longevity.
His research on animal learning and animal cognition The mentality of apes became a classic. He argued against Trial and Error Learning as the primary principle of learning. It only applied when the structure of the problem was hidden from the problem solver. Meaningful problem solving took place when the organism could see the structural relations among the parts and restructured the parts using meaningful relationships between the parts and the whole.
His brightest chimp, Sultan, became very well known. Sultan piled boxes on one another to reach a suspended banana. He also put two sticks together to use as a rake to bring food within reach of his cage. In these problems the parts were brought together to create a useable structure.
Köhler was the major early articulator of Gestalt Principles of Perceptual organization. These include:
Figure-ground, reversible figures, apparent motion, common fate, proximity, similarity, good continuation, closure, and illusions. All were explained by the Law of Pragnanz: "good Gestalt" components fit together.
This work set the stage for much of the current work in pattern recognition and scene perception.
Links to Köhler's Gestalt Psychology Today
An old pseudoproblem

Student of Stumpf.
Lewin was primarily known for his research in social psychology, and especially on Small Group behavior, and on principles of Leadership.
Action research during WW II. This was research that resulted in changing behavior of the real-life participants in appropriate ways.
Lewin discussed structural relations between parts and wholes with regard to social and physical environments. psychological and environmental fields and life space.
Lewin suggested that organisms either liked or disliked many objects and events in the world. He analyzed this by saying that they had a positive or negative valence. These valences played an important role in behavior.
Lewin was one of the first researchers on small groups and the study of group dynamics. He studied leadership styles and conditions of success.
Lewin studied conflict using approach and avoidance gradients similar to those of Neal Miller.

Specific contributions of Lewin include:
The Zeigarnik effect gives evidence for dynamic processes in behavioral organization. A person is more likely to remember an incompleted task than one that she has finished to her satisfaction.
The Von Restorff effect shows the role of context. Items in a list that are different from the others in some systematic way are easier to remember. e.g. a number among letters or a letter among numbers; a word among nonwords or a nonword among words; an upper case word among lower case ones.
Some of Lewin's research on animal behavior showed that animals often learned better when the things to be learned were more distinctive.