The brain uses a number of automatic mechanisms to determine what information to retain. Everything else naturally fades away.
The brain's overriding principle, given to it from millions of years of evolution, is to retain whatever is likely to be useful later for long-term survival. Since the future utility of information is impossible to predict, the brain uses a number of heuristics that have been honed over the millenia.
Here are some of the most well studied:
Repetition -- Things that happen repeatedly are either highly significant or irrelevant. However even if they are irrelevant -- like the background noise that you tune out -- they must be identified so that they can be removed from perception. When studying for a test, students often use repetition to activate the brain's importance circuits.
Primacy and recency -- Things that happened first are often more important because they predict what comes later. And things that happened most recently are often the most relevant because they are closest to the present. Things in the middle tend to get forgotten. This is why so many presentations start and end with an overview of the key points.
Surprise -- Anything that is unusual stands out. This can include an uncanny coincidence or an event that led to something unpredicted. An entertaining science teacher will ask students to guess what will happen and then show that the opposite happens. Setting up the experience of surprise increases retention. If you are thinking of calling someone and the phone rings and it's them, you will remember that for a long time because the coincidence is so unusual. Since you forget all the times you thought of calling without the phone ringing, you are left with the false impression that the coincidence was special.
Emotional impact -- Emotions are one of the ways the brain prioritizes perception and action. Emotions are a way of assessing and categorizing situations according to their role in our instinctual survival program. A moment correlated with a strong emotional state will be retained for a long time, which is why, for example leading up to a car accident, people have the memory of time slowing down and noticing every detail. Time didn't actually slow down -- it's just that a lot of detail got recorded and so the event is remembered this way.
Leads to positive or negative outcome -- The systems in the brain that learn behaviors and habits are especially tuned to the eventual outcome of an action or perception. This is why addictive activities such as gambling can be so tenacious. With a slot machine, most of the time nothing happens, but sometimes the bells ring and the sign flashes "winner!". Each dose of "reward" ensures that more quarters go into the machine. Addictive drugs like nicotine and cocaine activate reward circuits directly, causing everything that led up to taking the drug to be given automatic priority by the brain. Dopamine is a neurotransmitter central to signaling reward and activating procedural memory formation. When something leads to a strongly unpleasant outcome, emotional circuits label the preceding events as fearful.