One might imagine that 60+ years of development must have produced large gains, but chemical rocket performance is fundamentally limited by the amount of energy in the chemical fuels, and the 1960s engines were already getting at least 2/3 of the maximum theoretically possible performance (see comparison table below). //

The usual primary metric is specific impulse.

But specific impulse is a somewhat unintuitive quantity to understand, so let's start with effective exhaust velocity, which is the average speed of an exhaust particle (in the backward direction). For example, the Rocketdyne F-1 engines used in the first stage of the Saturn V (the Apollo rocket) have an effective exhaust velocity of 2.58 km/s at sea level.

What does 2.58 km/s mean in terms of rocket performance? It means if you build a rocket whose weight is about 63% fuel, and you fire the engine in deep space until the fuel runs out, the rocket will now be going 2.58 km/s faster in whatever direction it was pointing: //

So, what is change in velocity, Δv, good for? In the solar system there are two main uses for Δv: launching from the surface to achieve orbit, and transferring from one orbit to another. The article Delta-v budget has some examples, but the most relevant to Apollo is the Δv to get into low Earth orbit from a sea level launch, which is (very roughly) around 10 km/s. That breaks down as about 8 km/s of required velocity to stay in orbit (any slower and you'll come back down) and 2 km/s spent lifting the rocket against gravity and pushing through the air on the way up. //

So let's take a quick comparison of ve for the F-1 and the SpaceX Merlin engine. This is a relatively fair comparison because both burn RP-1 (refined kerosene) and liquid oxygen in a gas-generator cycle. These characteristics are good for a first stage due to high energy density per unit volume and high thrust, although other fuels have better ve

F-1 2.58 km/s (sea level)
Merlin 2.77 km/s (sea level)
F-1 2.98 km/s (vacuum) 65% of max
Merlin 3.05 km/s (vacuum) 66% of max
Theoretical max 4.61 km/s (vacuum)

The theoretical maximum is based on the total chemical energy in the fuel. //

Finally then, what is specific impulse? It's obtained from ve

by dividing by the gravitational acceleration on Earth:

Isp=veg

where g is usually standard gravity, or about 9.81ms2. The resulting quantity has units of seconds. For example, for the F-1 at sea level, Isp=263s

What is the physical significance of Isp?

Well, consider our rocket from before with 63% fuel by mass. Suppose we start the rocket while it is sitting on the pad, let it just barely lift off, then hover just off the pad until it runs out of fuel (this assumes we can arbitrarily throttle the engine without affecting its performance, which is not realistic, but ignore that). Isp is how long it will hover. That is because, for every second of hovering, we consume 9.81 m/s of Δv in order to overcome gravitational acceleration accumulated during that second. After Isp seconds, all of our Δv is gone.