Rate of change is, at it's simplest, how quickly something is changing. You can define any time period you like.
For example speed is the rate at which distance is changing in a given time, you can use the NY marathon as a rate of change.
Calculus comes into it because we can make the amount of time smaller and smaller to get a more accurate rate for that given part - if, for example I climb 3 flights of stairs in 1 minute the rate of change is 1 flight of stairs per minute. However, I could have dashed up 2 flights in 30s, rested for 10s and then took 20s up the last which means the average wouldn't work.
By decreasing the interval time a more accurate picture is obtained at that point. Calculus comes into it when there is an infintesimally small change