The fundamental components make up all other components and items.
Torch[]
Torches are used to hold sand, gravel and attachable blocks. When torches are washed away by liquids, or the block holding them falls/is destroyed, they drop as items, allowing sand/gravel to collapse, and attached blocks to drop, with the exception of Carpets when the torch is washed away by a liquid. This makes it useful for many applications.
Carpet[]
Carpets are used for mostly the same purposes, but it has some differences. Carpets can only be placed on top and only on top of any block besides air. This makes it useful for "dry-out" detectors, by having Carpets to be stacked where the liquid will be.
Gravity-affected blocks[]
Sand and gravel have ability to fall when there is no support. Any block except fire, air, water and lava can hold a block of sand or gravel, but only solid blocks can stop them. They will just fall through other blocks. If a solid block has a non-solid one, they will break on landing. This makes them very useful for a wide range of applications.
Sand[]
Aside from use as a falling block, sand is also used to hold cacti.
Cactus[]
Cacti can damage entities (e.g. destroy minecarts or hurting mobs), and is destroyed by a block appearing orthogonally next to it.
Cactus is used in "greenstone" wiring.
Liquids[]
Liquids can wash out some non-solid blocks (e.g. torches).
Water[]
Water can cool lava into cobblestone or obsidian, as well as safely push entities.
Water flows at speed of 4 blocks per second.
Lava[]
Lava can burn flammable blocks, ignite TNT and damage entities.
Lava flows at speed of 1/30 blocks per tick in the Overworld.
Sign[]
Signs can be attached to most blocks, including other signs. The latter fact allows for instant "circuitry" of any length.
Flammable blocks[]
Flammable blocks after being burned can let gravity-obeying blocks to fall or attached blocks to vanish.
Rail[]
Rails are used to guide minecarts.
Powered rail[]
Powered rails are used to propel minecarts.
In Pocket Edition, powered rails are always active. If you want to build redstone alternatives using minecarts in other editions, you may want to power them.
Minecart[]
Minecarts are destroyed by cacti or lava, letting what rides them to fall, perhaps onto a piece of farmland.
Farmland[]
Mobs falling onto farmland with seeds on it will trample farmland, which can update BUD switches.