The idea most famously associated with his name, and one that has an interesting history, is a distinction between r-selection and K-selection. In population ecology, r is the intrinsic rate of population growth; it represents the exponent describing the rate of population growth. Looking at an individual, it represents fitness, and works out to something like one less than half the expected number of surviving offspring per individual. "Something like" because we're ignoring any effect of density dependence. Density dependence occurs when individuals compete for scare resources, which results in mortality.
When resources are limited, there is some finite number of individuals who can survive in a population, a number usually represented by the variable K, for carrying capacity. Given a population growing according to certain well-understood models of birth and death, an ecologist can calculate the carrying capacity. When a population is at carrying capacity, the average number of offspring per parent will be 1 (each parent just replaces itself, on average); when the population is very far from equilibrium, the expected number of offspring will be close to r.
G. Evelyn Hutchinson and his student Robert MacArthur led the way to developing this equilibrium model into a mathematically elegant vision of ecology in the 1950s through the '70s, when Pianka was studying under MacArthur. In this view of the world, species interacting in communities were all at or near their equilibrium, and any disturbance was followed by a rapid return to equilibrium. Population levels of competitors and predators were in a fairly tight balance, controlled by the elegant equilibrium equations derived mathematically.
In thinking about this equilibrium ecosystem, Pianka argued that there were two basic "strategies" that an organism or species might find itself pursuing. When the future was unpredictable, or when resources were effectively infinite, selective pressures would act to increase r, raising the number of offspring and reducing the parental investment in each child. This is called r-selection. As population levels rise and resources become scarce, it's better to produce a small number of the best offspring possible, investing as much effort as possible to produce children who will be able to out-compete other offspring. In this case, survival is nearly zero-sum because the population is close to K. Of course, better competitors are likely to be more efficient, thus raising K. This is K-selection. Obviously, most species would occur in some balance between extremes.
Why would r-selection persist? Pianka argued that populations in environments that change rapidly relative to the rate of reproduction are likely to remain r-selected. Insects that reproduce more than once per year are leaving offspring which would face a radically different selective environment than their parents or grandparents because of annual or even monthly variation, while an elephant's life averages out changes in the environment that happen over decades.
Having r-selection or K-selection requires that all of these populations are rapidly returning to near-equilibrium. When the effects of r- and K-selection were tested in the field and in the lab, the predictions tended to fail. People still refer to r-selection and K-selection, but mostly as short-hand for discussing short-lived species which produce copious offspring or long-lived species which produce few offspring and invest in parental care. The formalism of the selection was wrong.
The problem, in large part, is that most populations are probably not in or near equilibrium. What the equilibrium approach tended not to consider was that it takes time to achieve equilibrium, and in a community with numerous interacting species and regular interruptions by weather, disease and other random events, it can take enormous stretches of time to get anywhere near the theoretical equilibrium. If the time between random events is shorter than the time it takes to recover from the event, you never get back to equilibrium.
On some level, it is this older worldview that is driving Pianka's concerns. It's a view in which population levels, whether of humans or of other species, are tightly regulated by competition within and among species. Humans moved past K, and there is an inevitable population crash that will result from that. This is a difficult claim to accept for reasons I described earlier. What's easier to accept is that we've caused dramatic changes which will take longer for the species to sort out than they'll have time to attempt it. What consequences that will have for our world and our future as a species remains unclear. I think Pianka is right that it's a sort of arrogance to continue on that course, and only to worry if human populations are at risk.