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The Doomsday argument
This article introduces the Doomsday argument (DA) in four alternative ways:

For a description of the DA without mathematics see the analogy-to-cricket section.
For a very simplified numerical example, partially based on Korb’s refutation[1] see the two-case section.
For a general overview, with numerical examples see the next section.
For Gott's mathematical development of the Bayesian argument (using simple calculus) see the Vague Prior section.
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Numerical estimate of Doomsday
Let us imagine our fractional position f = n/N along the chronological list of all the humans who will ever be born, where:

n is our absolute position from the beginning of the list.
N is the total number of humans.
The Copernican principle suggests that we are equally likely (along with the other N-1 humans) to find ourselves at any position n, so our fractional position f is uniformly distributed on the interval (0,1] prior to learning our absolute position.

Let us further assume that our fractional position f is uniformly distributed on (0,1] even after we learn of our absolute position n. This is equivalent to the assumption that we have no prior information about the total number of humans, N.

Now, we can say with 95% confidence that f = n/N is within the interval (0.05,1]. In other words we are 95% certain that we are within the last 95% of all the humans ever to be born. Given our absolute position n, this implies an upper bound for N obtained by rearranging

n / N > 0.05
to give

N < 20n.
If we assume that 60 billion humans have been born so far (Leslie's figure) then we can say with 95% confidence that the total number of humans, N, will be less than 20·60 = 1200 billion.

Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, one can calculate how long it will take for the remaining 1140 billion humans to be born. The argument predicts, with 95% confidence, that humanity will disappear within 9120 years. Depending on your projection of world population in the forthcoming centuries, your estimates might vary, but the main point of the argument is that we are likely to disappear rather soon.

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Remarks

The Wheel of fortune, 1510: Contrasted with Sapientia (wisdom) the blindfolded Goddess Fortuna and the Tarot wheel symbolize the Medieval belief in impermanence, representing the principle: This too shall pass.The step that converts N into an extinction time depends upon a finite human lifespan. If immortality becomes common, and the birth rate drops to zero, N will never be reached1.
The total number of humans born so far may depend on one's definition of "human".
By counting the number of human consciousnesses as states, the Doomsday argument (DA) imputes a special value to the human mind that rejects the Copernican tradition's mediocrity principle.
A precise formulation of the DA requires the Bayesian interpretation of probability, which is widely, if not universally, accepted.
Even among Bayesians some of the assumptions of the argument's logic would not be acceptable; for instance, the fact that it is applied to a temporal phenomenon (how long something lasts) means that N's distribution simultaneously represents an "aleatory probability" (as a future event), and an "epistemic probability" (as a decided value about which we are uncertain).
The U(0,1] f distribution is derived from two choices, which whilst being the default are also arbitrary:
The principle of indifference, so that it is as likely for any other randomly selected person to be born after you as before you.
The assumption of no 'prior' knowledge on the distribution of N.
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Simplification: two possible total number of humans
Assume for simplicity that the total number of humans who will ever be born is 60 billion (N1), or 6,000 billion (N2). If there is no prior knowledge of the position that a currently living individual, X, has in the history of humanity, we may instead compute how many humans were born before X, and arrive at (say) 59,854,795,447, which would roughly place X amongst the first 60 billion humans who have ever lived.

Now, if we assume that the number of humans who will ever be born equals N1, the probability that X is amongst the first 60 billion humans who have ever lived is of course 100%. However, if the number of humans who will ever be born equals N2, then the probability that X is amongst the first 60 billon humans who have ever lived is only 1%, such that the total number of humans who will ever be born is more likely to be much more closer to 60 billion than to 6,000 billion. In essence the DA therefore suggests that human extinction is more likely to occur sooner rather than later.

It is possible to sum the probabilities for each value of N and therefore to compute a statistical 'confidence limit' on N. For example, taking the numbers above, it is 99% certain that N is smaller that 6,000 billion.

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What the argument is not
The Doomsday argument (DA) does not say that humanity cannot or will not exist indefinitely. It does not put any upper limit on the number of humans that will ever exist, nor provide a date for when humanity will become extinct.

An abbreviated form of the argument does make these claims, by confusing probability with certainty. However, the actual DA's conclusion is:

There is a 95% chance of extinction within 9120 years.
The DA gives a 5% chance that humans will still be thriving circa 11125 AD. (These dates are based on the assumptions above; the precise numbers vary among specific Doomsday arguments.)