Concrete PSI is the compressive strength a mix is designed and specified to reach, in pounds per square inch, normally measured at 28 days, which is what the number in 4,000 psi concrete means.
When you order 4,000 psi, the plant designs the mix to beat that number with room to spare, because real batches vary. Strength keeps climbing after 28 days. That's just the age the industry agreed to measure at.
How it's checked. On jobs that require testing, a technician takes a sample from the truck and casts cylinders (ASTM C31), which are cured and broken in a lab (ASTM C39). One strength test is the average of two 6 by 12 inch cylinders or three 4 by 8 inch ones. Under ACI 318, the concrete passes when every average of three consecutive tests meets the specified strength and no single test falls more than 500 psi below it (for mixes up to 5,000 psi).
What drives it. Mostly the water-cement ratio. Less water per pound of cement makes stronger concrete. Water added at the truck, poor curing and early freezing all cost strength the mix was designed to have. Entrained air costs some too, which the mix design accounts for.
What to order. The engineer's spec or your local code sets the minimum. Exterior slabs that freeze and see deicing salt generally need a higher strength, air entrainment and a lower water-cement ratio than interior slabs. Residential flatwork is commonly 3,000 to 4,000 psi, but check your code before you price it.
Higher strength costs more per yard, mostly because it takes more cement. Put the psi on the proposal and on the ready-mix order, so there's no argument about what was poured.