Extinction of species with extant descendants
You are right. Consider a population at some point back in time and ask how many of the individuals left descendants that are still alive today?, the answer will be between 1 and N (inclusive), where $N$ is the population size. We call this process coalescence

A tiny introduction to coalescence

Consider a population of constant size $N$ and to the previous generation and ask the question What is the probability of two random individuals to be siblings?. This probability is $\frac{1}{2N}$ (assuming no selection and diploid population). The probability of them not being siblings is $1-\frac{1}{2N}$. We can say that, with probability $\frac{1}{2N}$, two randomly individuals coalesce in the previous generation. What is the probability that 2 random individuals coalesce $t$ generations ago? It is the probability that they don't coalesce for $t-1$ generations and then coalesce, that is $P(t) = \left(1-\frac{1}{2N}\right)^{t-1} \frac{1}{2N}$. In words, the coalescent time follows a geometric distribution with parameters $t$ and $\frac{1}{2N}$. 

Because coalescent times are independent variables (and I won't prove it here), one can extend this to not only 2 individuals but $n$ individuals. There is therefore, a point in time where all individuals of the modern day population coalesce into only one individual. At this time, this one individual was not alone in the population of course.

Coalescent theory: Usages and book recommendation

The simple calculations I just made are the very basis of coalescent theory. Coalescent theory is a very powerful set of mathematical tools that one can use to make detailed predictions of the genetic patterns to expect in a population. Statistical test such as Tajima D test is based on results from coalescent theory. Coalescence theory can be extended (through approximations) to structured population (structured coalescent theory), non-neutrality (selection) and recombination.

coalescent theory: an introduction (by John Wakeley) is a good book on the subject. However, it requires from its reader some basic knowledge in mathematics and problem some knowledge in population genetics.

What is the percentage of people living in England in 1500 AD whose lineage is still alive? is a recent post on Biology.SE that is relevant to the kind of question that could be answered with coalescent theory.