Grass is one of the three Starter types, forming a perfectly triangular relationship with Fire and Water.
Many Grass types are based on real-world plants and fungi; many such Pokémon also belong to the Plant egg group. Several Grass types are paired with the Poison type, reflecting the toxicity of several plants towards mainly humans. It is worth noting that the first five Grass-type starter lines are based on prehistoric or current species of reptile.
Most Grass-types can learn "seed", "spore" or "powder" moves that inflict status (usually Poison, Paralysis or Sleep), including a unique type of status caused by Leech Seed; thus, many Grass-types employ a passive style of play during battle. In terms of direct attacks, Grass-types drain with moves like Giga Drain, or attack with leaves, vines, petals, wood or sunlight.
They're strong against the Ground type because plants absorb minerals from it and open through the ground using their roots.
They're strong against the Rock type because plants can crack stones and rocks using their roots.
They're strong against the Water type because plants absorb water for their own benefit.
They're weak against the Bug type because insects eat plants.
They're weak against the Fire type because fire can easily burn plants.
They're weak against the Flying type because many birds eat parts of plants (like fruits and leaves), as well as strong winds being able to uproot plants.
They're weak against the Ice type because plants can't resist frost, and die most of the time as a result.
They're weak against the Poison type because poison is used to exterminate unwanted herbs. It could also be said that Poison represents pollution and Grass represents nature, and nature is damaged thanks to pollution.
They're resistant against the Electric type because plants are made of materials that don't conduct electricity, like wood. Also, plants divert electricity to the ground.
The Steel type resists them because plants can't degrade metals. It might also come from the fact that machines (like weed-wackers and lawnmowers) are often used to kill plants.
The Dragon type resists them because in ancient mythology, dragons were immune to forces of nature (Grass, Water, Electric and Fire).
Trivia
All Grass-type starter Pokémon until Generation V resembled a lizard or dinosaur of some sort. Chespin and Rowlet broke the trend.
The Grass-type is tied with the Rock-type for the most weaknesses (5).
However, if both are combined, the weakness outcome only comes out as 4.
Bulbasaur and Rowlet are the only basic stage duel-typed starter Pokémon. Curiously, they both are Grass-type starters.
All of the final evolutions of the Grass type starter Pokémon resemble an extinct animal: