Range and interquartile range
Two data sets share the same middle. How do you say which one is more spread out, without letting one strange value decide?
Where you are 2 inside
What this area is about
A center says where data sit; a spread says how far they wander. The simplest spread is the range, largest value minus smallest. One extreme value can stretch it a long way, so a sturdier measure looks only at the middle half of the data: the interquartile range, \text{IQR} = Q_3 - Q_1. Both come from cutting sorted data at fixed places.
How these fit together
Definitions. Quartiles and the five number summary cuts sorted data into four parts of about equal size at Q_1, the median and Q_3. With the minimum and maximum these make the five-number summary, and the range and the IQR are read straight off it.
A procedure. Percentiles generalizes the cuts from quarters to any percent: where one value sits among the others, and which value sits at a given percent.
A path through it
Quartiles first, then percentiles, since Q_1 and Q_3 are the 25th and 75th percentiles (the two methods can differ slightly for some list sizes).
The five-number summary is what a box plot draws, and the IQR powers the 1.5\,\text{IQR} outlier rule. Percentiles return in normal calculations.