My CFI said it gets 100 hr inspections, but that it has almost 10,000 hours on it. When we got in, the Hobbs meter said 6170. Is the Hobbs meter recording hours on the engine? What is the difference between the Hobbs meter and the "almost 10,000 hours" my CFI mentioned?
A Hobbs meter is essentially an electric clock that runs whenever power is applied to it at the rate of one hour per hour. Its primary application is billing for use of the airplane at an hourly rate. Usually, it is powered directly through the electrical master switch, although many flight schools/FBO's wire it through an oil pressure sensing switch so the renter cannot "beat the clock" for billing by turning off the master in flight.
A recording tachometer is essentially an odometer operated by engine revolutions (rather than wheel revolutions as on your car). It is geared to show one hour of time rolling over after some particular number of engine revolutions, typically around 2400-2500. Thus, it rolls over about one hour for every hour in cruise flight, but slower during approach and pattern work, and very much slower when idling on the ground. Its primary designed function is measure engine wear for maintenance purposes, but it is usually used also for all other aircraft wear maintenance purposes (such as 100-hour inspections, wear-based Airworthiness Directives, component overhauls, etc).
In normal aircraft use, it is typical to see the tach time about 80-85% of Hobbs time on any given flight, with the percentage higher when flights involve long trips at high cruise, and lower when there's a lot of primary training at lower power settings (traffic pattern flying, slow flight, etc).
In either case, the device may be replaced during the life of the aircraft. In some cases, the new device is rolled forward to match the reading of the removed unit, but in others it is just installed at zero and the aircraft logbooks are used to track how much time it had at device installation for future reference of total time. Thus, it is entirely possible for an aircraft with more than 10,000 hours total airframe time to have a Hobbs meter installed long after the aircraft was built reading 6171 hours.
I read online somewhere that the typical lifespan of a 172 is around 10,000 hours. Is that true? Obviously there are a lot of factors that play into that, but generally speaking?
There are plenty of 172's out there in the training fleet with considerably more than 10,000 hours that are still going strong. OTOH, there are plenty of abused/neglected 172's which are basket cases after half that. It's just not possible to give any specific number as being the "typical lifespan" of a 172 without knowing more about its use and care.
