Modern day land plants --- as well as modern day water-based plants --- are believed to have evolved from algae. Algae are a broad classification that includes most microscopic, photosynthetic single-celled organisms; some algae are prokaryotic protists --- meaning that have a nucleus that contains their DNA --- and some are eukaryotic bacteria --- which lack a nucleus. Algae are the most common ocean-based organism, and they are one of the main providers of oxygen for the world's ecosystem. Several of the important organelles found in modern day plants --- such as chloroplast, which is where photosynthesis takes place --- are found in algae.
There is no clear fossil record that connects algae to land-based plants; however, the first land-based plants are believed to have evolved around 450 million years ago. These are the bryophyta, which still exist today in the form of liverworts and mosses. Bryophyta have no system for transporting water within their internal structure, thus they must situation directly over a water source, and as a result are limited in how large they can grow.
One of the most important changes to occur in land plants was the evolution of a vascular system. The vascular system is connective tissue that allows for fluids to be transported amongst various areas of the plant; as a result, plants developed specialized parts, such as leaves for photosynthesis and roots for absorbing water. Vascular plants are believed to have first evolved around 420 million years ago; modern day seedless vascular plants are classified in the Pteridophyta domain, with the most common species being ferns.
The next important advancement in the evolution of land-based plants was seeds, which gave them the ability to spread over great distances. The original seed-based plants were gymnosperms, which are believed to have evolved around 350 million years ago. Modern day examples of gymnosperms include conifer trees.
The next advancement in seeds was the evolution of angiosperms, which was around 300 million years ago. Angiosperms use flowers to spread their seed, which is protected in a special casing called "pollen."