The wood part is called the 'blade' in English and there are MANY different brands and models. They come in either 5 ply or 7 ply, where it might be full (5 or 7) wood or 5 wood plies + 2 thin carbon wafers. I use a carbon blade (Viscaria) - they are more stiff with less vibration, or feedback, but generally have a higher speed ceiling and larger sweet spot at the expense of a little bit of control at lower speeds (gears). The other component of a setup is the rubbers, you can have one black rubber and one red rubber (although new colours are soon being made an official part of the sport) - generally they're split into two categories "tensors" and "tacky" or "chinese" rubbers and recently "hybrid" rubbers. Rubbers are comprised of a sponge and a top-sheet - which manufacturers can customise to produce different playing characteristics. Long story short: tensor rubbers historically were favoured by Europeans (on both sides, whereas Chinese would use a hard tacky rubber on forehand + softer tensor backhand) and are more elastic with built in catapult effect. They require less physical effort to produce good spin and speed. Tacky Chinese rubbers are very hard with a very tacky surface, the harder sponge makes them easier to control at low gears (close to the net for short game) and have great top-end spin as well as spin on serves. They require more physical exertion and efficient use of the body to use effectively (although there are also substances called "booster" which we apply to make hard tacky rubbers come alive). Hybrids are a sort of mixture of the two that have become popular with the new 40mm plastic ball and can be used on either side. Typically you will use a harder rubber on the forehand than on the backhand, but it's up to personal preference.

The rubber and the wood (or other material) itself, that is enough for people to delve deep into. Different rubbers on different sides.

The first page of results for "table tennis ej" on Google has all the answers. I had no clue about it either.