What is a typical retirement pension plan for professors in the United States?
AFAIK, all pensions are calculated basically the same, and I'll use Alabama, Tennessee, and Georgia as examples since I'm familiar with them.  In the public system, these are often pooled between all universities and colleges and often with the K-12 system or all public employees.

You have three variables: the eligible salary, the number of years worked, and the multiplier.

The eligible salary is calculated through a variety of methods, but tends to be the average of the highest X years of work over the last Y years.  For example, in Alabama, it's the average of the highest 3 years of the last 10 years.  In Tennessee, it's the average of the highest consecutive 5 years.  In Georgia, it's the average of the two highest consecutive years.

The multiplier is probably the single most important variable, though.  My father, for instance, worked in Alabama, with a 2.0125 multiple.  Let's say for instance his eligible salary was 100k to make the math easy.  If he retired with 10 years of service, he would receive ~20k per year.  With 20 years of service, ~40k, etc.  For me in Tennessee, my multiplier is only 1.5.  If when I retire I have an eligible salary of 100k, with 10 years of service, I'll only make 15k.  With 40 years, I'll make 50k.  (Yeah, my dad got a much better deal, and that's before we talk about the DROP program).  Georgia, for reference, has a 2.0 multiplier.

A minor difference thing to consider are time to vesting (Tennessee is only 5, but Alabama and Georgia are 10), but over an entire career, those don't make much of a difference for overall payout.

Because the ways to calculate the eligible salaries (highest two years really helps folks in departments with rotating chairs), I'm not sure how well one can generalize about the pensions.  One thing for sure is that they tend to be quite decent since you also collect social security benefits.  Anecdotally, for instance, between his pension with 30 years worked, social security and differing retirement tax rules, my dad takes home more now than when he worked.  Plus in academia we tend to be able to work well into old age which really boosts our pension income.

Based on what I've read, private pension funds work the same way, just with ever so slightly different variables.