Plants grow using things called meristems which are layers of cells capable of rapid cell division. There are two types of meristems: (1) the shoot apical meristem where the plant grows taller and gives rise to new leaves or branches and (2) root apical meristems where the root grows longer. Plant growth can also be primary or secondary. Primary plant growth allows the plant to increase its length while secondary plant growth allows the plant to increase its girth.
You may be familiar with in people there's these kind of cells called stem cells, the cells that can do rapid cell division and aren't necessarily already specialized in what they can do and it's an area that people are really interested in figuring out so that we can do things like repair our skin and stuff like that. Well with plants they have similar layers of cells but instead of being called stem cells they're called meristems and this is how plants are able to achieve their growth. So these kinds of or layers of cells are further subdivided into the protoderm, ground meristem and procambium. The protoderm you might be able to figure out from that root word 'derm' there is what gives a rise ultimately to the outer layers of the plant. The ground meristem gives rise to what's known as the ground tissue which is often involved in things like photosynthesis or storage. And procambium which is what gives rise to the vascular tissue things like xylem and phloem, occasionally you'll see when these start to specialize you'll see things mentioned like vascular cambium which is tissue that gives rise only to the vascular tissue and cork cambium which is what gives rise to the cork, the bark of some kinds of plants. Now at either end of a plant, at the tip of their shoot, the growing part above the ground and at the tip of the root you'll have the shoot apical meristem and root apical meristem and this is where the plant is able to add new cells as it grows. Now this is what in the plant, the shoot apical meristem is how it gives rise to new branches as that shoot grows the cells up here start to grow and develop into new leaves, they may grow out laterally to extend into other branches and a very similar process how the root grows. Now most plants can do what is called primary growth, that's just getting longer at either end and elongating their lateral branches. Secondary growth is something found only in some plants and that's where it's able to add additional layers of cells going outwards and this is found in things like Oak trees or Redwood trees and that's how they get bigger and bigger and bigger, whereas things like bamboo or corn, they typically can't do that secondary growth. If you've ever opened up a bamboo stem you'll actually find it's pretty much hollow on the inside. So that's a difference in how some plants can grow.