The rank between the family and the flower
A genus sits one step below the family and one step above the individual species. It is the first half of a scientific name, so in Rosa gallica the genus is Rosa, and everything sharing that first word is a close relative in a way that things merely sharing a family are not.
Only five genera have pages here, and the reason is worth explaining because it looks like an omission. The flowers in this guide span 102 genera, and for 97 of them the genus holds exactly one flower whose common name is the same word as the genus. A page about Alstroemeria the genus would say what the alstroemeria page already says, so it would compete with it rather than add to it.
These five are the exceptions. Each one holds several flowers that are sold under different names, at different prices, for different purposes, and each has something to say that belongs to none of those pages on its own. That is the whole test, and if a future flower makes a sixth genus hold more than one, it will earn a page on the same grounds.