Why is it that, even if we were going to immediately agree that every women will not have more than 2.1 children on average, the world population would continue to grow for another 60 years? I've learned that a TFR of 2.1 means the population is at 0 growth rate. I guessed the answer was the population-lag effect, but since we are assuming that ALL women immediately have only 2.1 children on average, wouldn't there be a 0 population growth immediately and not after 60 years? Thanks in advance!
As you mention, the population-lag effect is responsible for this. From The Wikipedia article on TFR: http://en.wikipedia.org/wiki/Total_fertility_rate#Population-lag_effect
Thus, even if the current TFR implies long-term stability, the recent history of TFR values will continue to affect birth rates, and thus population growth/decrease, in the future.
Imagine that you have a stable population, with static birth and death rates. Now imagine that the birth rate during a single year for some reason doubles before dropping back to normal the next year, thus transiently increasing TFR. Now, it is easy to see that when the newly born, larger-than-normal generation reaches reproductive age, the population will increase again because the birth rate will increase in proportion with the size of the generation, while the death rate is unaffected until the enlarged generation reaches old age and starts dying off. Later, the population will decrease to its long-term stable level, which will be larger than before the TFR spike.