A base in general terms is just a correction period following a stock making a new high (there are exceptions, such as a bottoming base, formed after a prolonged downtrend, but even that will have a relative high serving as a pivot point).
When counting bases, the stock must run up at least 20% from its breakout in order to form another base to increment the count. A base on base is a formation where the stock breaks out, but then corrects again before it has gained 20%. This is all consided by O'Neil to be a single base.
