It appears to be primarily inhabited by bug-type Pokemon including the Caterpie, Weedle and Venonat lines. However there are other Pokemon who inhabit the forest, including Kangaskhan and Pikachu. Red's Pika and Yellow's Chuchu both originally came from here.
There is at least one river running through Viridian Forest, suggesting a plentiful amount of water-type Pokemon, such as Seaking and Seadra.
Mew also appeared in Viridian Forest.
Viridian Forest has been the setting for several events in the series. Like many areas in Kanto its ecology has been thrown out of balance by pollution and experimentation by humans, especially Team Rocket.
It is also said that once every few years a child is born with the mysterious powers of the forest. These powers tend to depend on the person in question but typically they allow the possessor to heal the wounds and read the thoughts of Pokemon. Yellow and Lance have both displayed these powers. Additionally while Yellow appears to be able to telepathically control inanimate objects and raise her Pokemon's levels through sheer willpower, Lance has the ability to phsyically control his Pokemon, especially if they are Dragon-type or related Pokemon.
Giovanni is also from Viridian Forest although he has yet to display any of its powers. Nonetheless it is presumbed he does at least have some mysterious power as he was recognized by both Yellow and Lance to be "one of [them]."