Question: I'm reading a paper on jet engines that repeatedly mentions the stagnation pressure and temperature. What does this mean? Is it the point where the jet engine stalls in performance/efficiency/thrust, etc?

I'm a bit confused. 99% of the information on stagnation pressure is the static pressure, or the about stagnation point, or pressure where fluid velocity is zero.

Answer: 
[![enter image description here](https://i.stack.imgur.com/RtPh1.png)](https://i.stack.imgur.com/RtPh1.png)[Image source](https://www.researchgate.net/figure/Smoke-visualisation-and-flow-topology-of-the-airflow-around-a-cylinder_fig7_260045652)

Stagnation pressure is used as a synonym for total pressure. An object in airflow experiences pressure from the flow = dynamic pressure, indeed for incompressible flow proportional to the velocity squared.

In the pic above, on the right hand side, is an indication of the Stagnation Point. This is where the air streaming in does not flow around the cylinder but is stopped by the body contour. The flow stagnates, and all inflowing dynamic air pressure is converted into static pressure at this point.

Which is called the stagnation pressure.
The analytical formula for the stagnation pressure $p\_{st}$ is :

$$p\_{st} = p\_{\infty} + \frac{1}{2} \rho V^2$$

with $p\_∞$ static pressure, $\rho$ = air density, $V$ = air speed for incompressible flow.

The point where the fluid is at rest is the point where stagnation pressure = static pressure ($p\_{\infty} = p\_{st}$), because the kinetic energy of the fluid is 0.

For compressible flows you can calculate the ratio of Stagnation/Static pressure with the formula :

$$\frac{p\_{st}}{p\_∞} = {\left( 1 + \frac{\gamma - 1}{2}M^2\right)}^{\frac{\gamma}{\gamma - 1}} $$ , simply by replacing Mach=0 then you can see that the stagnation pressure = static.

Try understanding how pitot measurement for pressure works and try studying Anderson for Fundamental Aerodynamics.

Hope that helps