It will not damage your car if it is new enough, because newer cars are able to correct for the lower octane. Which car do you have?
You'll have reduced performance, and worse gas mileage because the car is going to reduce compression and whatnot.
Essentially it's trying to avoid engine knock due to Regular gas (87 octane) not being able to handle pressure as well as higher octane gas. It's more likely to detonate under pressure, which is bad for the engine.
In older cars, it would cause some serious damage if you used too low of an octane for too long. The newer ones can compensate for it though. But you end up paying for it with worse gas mileage, and a sluggish car.
1999 Golf GL 2.0 8v