Grass is one of the three Starter types, forming a perfectly triangular relationship with Fire and Water.
Many Grass types are based on plants and fungi, not necessarily grass (Pokémon such as Cacturne, which is a cactus, and Foongus, which is a mushroom). Many Grass-type Pokémon also belong to the Plant Egg Group. Several Grass types are paired with the Poison type (Pokémon such as Oddish, Venusaur and Bellsprout), reflecting the toxicity of several plants towards mainly humans. Most Grass-types are also simply animals with plant life attached to them (Pokémon such as Bulbasaur, Paras, Sceptile, Gogoat and Tropius). Some Grass-types are also based on mythical creatures (Pokémon such as Shiftry, which is based on a Tengu, and Tangela, which is based on Medusa).
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 HP from opponents 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). Also, an extremely powerful wind gust can cause plants to be uprooted.
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 herbicides like weedkiller are 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).
They resist their own types because plants send out oxygen to other plants for their benefits like photosynthesis.
Trivia
All Grass-type starter Pokémon until Generation V resembled a Reptile of some sort. Chespin, Rowlet and Grookey 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 dual-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: