Non vascular plants were some of the earliest plants life to evolve on earth. Formerly, there were two large categories of non-vascular plants: bryophytes and algae.
However, recent science has complicated this in two ways. Firstly, the informal grouping called bryophytes has been divided in to Bryophyta (mosses), Marchantiophyta (liverworts), and Anthocerotophyta (hornworts). Secondly, it has been discovered that the organisms scientists have been calling algae are not as closely-related as thought, so only a subset of algae species in the botanical clade Viridplantae (green plants) can properly be called non-vascular plants.
TL;DR (Too Long; Didn't Read)
Non-vascular plants include mosses, liverworts, hortworts and certain kinds of algae.
What are Non-Vascular Plants?
Non-vascular plants are those without xylem and phloem structures: these structures are the plant equivalents of veins and arteries. These unique plants are unable to absorb moisture through their roots and transport it throughout its structure like vascular plants do. Instead, moisture is absorbed by the plant through its top surface area. Because of this, they generally grow in damp areas, however, there are some that have adapted to life in dry areas.
Liverworts are non-vascular plants. Their shape resembles a liver. They are the most basic living plants ranging in size from 0.02 to over 8 inches in diameter. They are one of the first plants to evolve from a sea plant to a land plant 400 million years ago. Liverworts are included in the Hepatophyta division, which consists of about 8,500 different species that grow in locations all over the world, including the arctic and the tropics. There are approximately 60 different families of liverwort plants.
Liverwort plants commonly grow in moist habitats, but there are some types that can grow in dry sandy areas and rock outcroppings. There are approximately 60 different families of liverwort plants. They grow in two different forms, either leafy (which have a leaf-like appearance), or thallose (which grow in large flat sheets). Leafy liverworts can easily be mistaken for mosses.
Mosses typically range in size from less than 1/2 inch to 3 feet. In the fossil record, they date back around 360 million years. Mosses are an extremely diverse group of plants that includes about 10,000 species divided into 700 genera.
Mosses grow all over the world, mostly in moist, shady areas. However, there are also types of mosses that can live in dry sandy soil and on rocky outcroppings. Mosses are very useful for preventing soil erosion and breaking down earth beneath the topsoil, making it easier for other plants to utilize. Mosses grow in either an erect form or a prostrate form, and are classified as either sphagnum mosses, true mosses or lantern mosses.
Hornworts are so-called due to their long horn shaped sporophytes, which are the parts of the plant that produces spores. Sporophytes are located on the top of the plant, stay attached and grow for the entire life of the plant. They range in size from 1/2 to 3/4 inch. There are very few fossil records of hornwort plants, but is believed that they have been around for about 380 million years.
Hornwort plants grow in temperate and tropical regions all over the world. They can be found growing on tree trunks, along riverbanks and in other damp areas. They attach themselves to a substrate with tiny hairlike anchoring structures. Hornworts look similar to thallose liverworts, and can be hard to recognize without their sporophytes.
As mentioned, only certain algae can be called non-vascular plants. Most of these are green algae. There is, however, significant debate over how well algae fits in to this schema at all, given that different kinds of algae can be classified as plants, protists or cyanobacteria.
With this in mind, it's best to think of non-vascular plants as liverworts, mosses, hornworts and sometimes algae.