Wikipedia defines it as:
In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon “leaf” and táxis “arrangement”). The process has been thought to be governed by the plant in order to minimize entropy.
So there are three types of (spiral) plant growth, and the only difference between them is how large the angle is for the next e.g. leaf or petals (which are modified leaves). So if the angle is 90 degrees its named it is denoted whorled arrangement. If the growth occurs on the opposite side of the stem of the plant - angle 180 degress - it denoted distichous plant growth. The majority of plants 80% - which is called spiral growth and has an angle in radians of:
where $\Phi$ is the Golden Ratio. So this works out to the constant value of 137.5 degrees.
The current understanding has been that the plant tries to minimize entropy while growing a new bud (primordium) on the growing stem (called meristem). Earlier proposals where focusing more on the bud trying to grow where there was more room. But the most recent research points to a combination of biochemical hormone signalling (auxin) and mechanics of bud growth as a thin elastic shells (see the talk by Newell,Shipman and Chung).
There was a special issue of “Annals of Botany”, vol 107 issue 5 published 2011 and I will try to add some of the gist from that issue.
Phyllotaxis, the arrangement of a plant’s phylla (flowers, bracts, stickers) near its shoot apical meristem (SAM), has intrigued natural scientists for centuries. Even today, the reasons for the observed patterns and their special properties, the physical and chemical mechanisms which give rise to strikingly similar configurations in a wide variety of plants, the almost-constant golden divergence angle, the almost constant plastichrone ratio, the choices of parastichy numbers and the prevalence of Fibonacci sequences to which these numbers belong, are at best only partially understood. Our goals in this Addendum are:
A Newell, P Shipman, Z Chung, Talk in 2009 Phyllotaxis and patterns on plants
Phyllotaxis, Wikipedia.
P Ball OUP 2009, Shapes. Nature’s Patterns: a tapestry in three parts