The doomsday argument is an argument that it is likely that a significant portion of all humans to ever be born already have been born, because if the total number of future humans is vast compared to the number of humans so far, then it is an unlikely coincidence for a randomly selected human to be born so close to the beginning.
For example, let's compare the hypotheses that the total number of humans to ever be born is 200 billion, versus the hypothesis that it is 100,000 billion. It has been estimated1 that about 108 billion humans have been born so far. Let's round that to 100 billion for simplicity. If the number of humans who will ever live is 100,000 billion, then the probability of a randomly selected human being one of the first 100 billion is 0.1%. However, if there will only ever be 200 billion humans, then the probability that a randomly selected human being one of the first 100 billion is 50%, which is much higher. According to the doomsday argument, this is significant Bayesian evidence in favor of only about 200 billion humans ever being born relative to the hypothesis that 100,000 billion humans being born.
The doomsday argument relies on the self-sampling assumption (SSA), which says that an observer should reason as if they were randomly selected from the set of observers that actually exist. In contrast, the self-indication assumption (SIA) says that an observer should reason as if they were randomly selected from the set of all possible observers. According to SIA, you are more likely to exist at all in possible universes with more humans, which cancels out the evidence from the fact that your position in the sequence of human births is more likely in universes with fewer humans. Thus, a reasoner operating under SIA would not use the doomsday argument as evidence against large numbers of humans existing in the future.