Unlike Decartes, and his theory of 'I think therefore I am' in order to prove existence, Existentialists believe that mere thinking cannot prove existence. Actually doing things, having emotions and producing actions are the way to test existence. They believe we can choose how/what to think.
This image shows both a rabbit and a duck. We can choose to see either the rabbit or duck but cannot see both at the same time. This helps prove existentialist theories.
Existentialists claim that as children we are all born with an 'Existentialist Attitude' where we experience a series of emotions including confusion and disorientation. This proves our existence, simply because we can feel these emotions. This marks the 'starting point' in life. Most Existentialists claim that previous philosophical theories are too abstract to be applied to human experience. Existentialism became popular during the post-war period as individuality was trying to be re-injected into society.
Kirkegaard is considered to be the father of Existentialism.
Kirkegaard believed that each individual was responsible for their own actions and how they lived their lives, they had the option to choose what their meaning of life was for them. He explained that during life, we are all faced with 'Existentialist Obstacles' and it is our choice how we overcome them. These obstacles are despair, anxiety, absurdity, boredom and alienation. He was a huge believer of Individualism.
Fredrich Nietzsche
Nietzsche agreed with Kirkegaard and focused his work on human experience rather than mathematics and science. He claimed that these natural sciences are too observational to measure human experience. He was interested in the roll of making free choices and how that changes the nature of individuality.
Nietzsche had created a character called 'Ubermensch'. This was meant to represent an idolized individual. An individual who created his own values and decided when and how he excelled them.
Kafka
Kafka wrote a novel called 'The Trial'. In this an individual was sent to court for a crime they were not aware they did, they had no recollection of doing anything wrong. In the courtroom and outside, no one would tell him what crime he had committed. As a result, he drifted away from his family and friends and became alone. He experienced feelings of confusion. This is said to be 'Existentialist Attitude'. Kafka wanted to show how individuals can choose to overcome this.
The key defining point of Existentialism is that existence comes before anything else. What we learn comes next. Even though an individual exists, nothing exists that marks their character. The individual chooses who their character is. Existentialists describe the feeling of abandonment that people feel when they realise they make this decision on their own and that there are no set values within society that we should follow.
Heidegger
Published 'Being and Time' which was another way of explaining human existence. He claimed that we shouldn't consider natural sciences as they absolute truth much like Nietzsche. He claimed that we if start considering sciences as the truth, then this will diminish our confidence in our own intellect.
Husserl
Husserl believed that the meaning of life is subjective. He agreed with Kirkegaard that as humans we have the option to choose who we want to be and what values we follow. He was very much a believer of individualism. He disagreed with Moore's Utopia.