Most of the answers take this from a earth perspective, but they don't always go into what they would actually look like, so I'm going to suggest some very different possibilities.
Since there is little wind, the leaves are free to get as big as they want. I'm not a botanist, but I'd assume the size of leaves is somewhat related to how much wind an area gets. The more wind, the smaller the size of the leaves, since the wind would likely move the leaf out of alignment of the sun, getting less energy and being less productive.
Also, small leaves allow for gaps in coverage, in a world where the sun and wind moves things, that's not necessarily a bad thing, but when the sun is stationary and the leaves don't move much, gaps become much less productive.
I can imagine leaves being massively large, even to the point where it's only one gigantic leaf for a plant. Since this would be difficult to hold up with a single branch, I can see the plant producing multiple branches at various points across the underside of the leaf. Not only would this help in weight/balance management, but it would also provide multiple points for nutrients to flow. Even a plant understands distribution of work and avoiding a single point of failure.
These leaves become a canopy that soak up all the sun and work in competition of other plants.
They also serve as a repository for water. They effectively water themselves during times when rain isn't as prevalent. The leaves would likely have a way to drain themselves, if they get too much rain. This could be as simple as letting themselves rip then healing the tear, or it could be as complicated as sucking in more water when the strain on the branches gets to be too much, or it could be slits in the leaves that open and shut on command.
Because your plants grow large canopies that prevent most light from getting to the ground, the fruits become larger, so they can germinate and grow longer without the need of sunlight until they get to be a considerable size. These saplings may have adapted to pierce another plant's canopy leaf to get their own light and water. They might even cling to an existing plant, like a vine, to get tall enough to get to the sun, then expand their trunk which potentially chokes out the climbed plant.
Plants may end up sprouting on other plants more, sucking their nutrients since they don't have the sunlight to produce their own energy. Depending on how close these plants are, the dependent plants may branch out and leech off multiple plants at the same time.
BBeast has a good idea on how this would work, but the examples given are explicitly based on the idea of the sun being on the horizon, yet a significant portion of the globe wouldn't be that way. There's plenty of land where the sun would be considered "up", so the plants would point in whatever direction/angle they needed to in order to get more sunlight.
Also, older growths would tend to grow higher, so they don't have to compete with newer growth. There may be trees that snake along the ground, but even at the sunset ring, I'd imagine trees that are at around 45 or 60 degree angles, or trunks that have gnarly changes in direction depending on how their competition was during different growth periods.
I can definitely see these trunks shooting out roots at various points, but not just into the ground. They may also take root in other trees and plants, wherever they can gain a foothold to prevent falling over. Of course, this leads to the trees being leaned on needing to put out more roots for their own management. This could lead to a massive tangle of trunks and root systems that create a barrier to anything of any significant size, as literally everything would be woven into one big mass. Even old trees that died would potentially still be supported and supporting other plants. These dead trees may even sprout new plants that add to the mesh. Heck, living trees may end up having new growth coming from them.
You'd see more plants like water lilies, where it's again a big leaf. Seaweed would likely be more horizontal than vertical. Algae would likely choke out some places, since there wouldn't be wind to create waves to move the surface around.
This would be a drastic change of scenery. Instead of massive trees and other plants, you'd see moss, mold, lichen, and other small, ground hugging growths. Because the daylight would have a fairly well defined cutoff point, you'd likely see as drastic a change in the foliage, too. It'd go from big stuff in massive forests to little stuff in just yards/meters. While there might be some truly large plants at the edge that have managed to come from this night side, they would be few and far between.
All the flora on this side would be relying on little to no light. Just a tiny bit from stars, some fraction from atmospheric refraction and clouds, and a small bit more from any satellites. Anything that can grow in Earth caves or generally underground without light would have an analogue here. There may even be large colonies of bacteria that grow openly.
Algae would likely dominate the water plant life, yet there probably wouldn't be much. Any water based plant live would be at the surface. Not enough light would penetrate to any depth for many plants to come from the bottom, except at the very shallow edges.
Because of the major differences in plant life, there would be major differences in animal life. Animals would flourish on the bright side. Since there's no night and day, they would likely be just as varied as daytime animals here on earth. Some would be fast predators that rarely sleep, some would be pack animals that have guards during rest periods, some would be slow and armored, some would burrow into the ground, some would hide in the foliage, some would make nests, some would swing from the branches, some would fly, some would walk the ground, some would be large, some are small, and all would be geared towards what area of the world they generally lived on.
The dense meshes of the outer ring would likely be dominated by smaller animals that can quickly dart through the mess to avoid predators or follow their prey. Animals nearer the sun "pole" would tend to be larger, since the canopies could be directly overhead and prevent a large amount of competition, allowing for more room between plants.
Any fish would likely eat a lot of algae, except for the predators. Even predators might eat some plants, due to it being so available even deeper. Because the sun never moves, there would be more plant life in the water near the sun "pole", but as the bodies of water get to certain point, the angle of incident and refraction would cut off the sun before the sunset ring, so plant life in those locations would drop off sooner.