Ritual: Requires a specific spell and monster
Fusion: Requires a fusion spell and listed monsters.
Synchro: Requires "Tuner" type monsters (A specific type, like how Gemini, Toon, spirit and Union monsters are their own type) and non tuners. Sometimes the tuners/nontuners have to be of a specific type like machine. The sum of the levels of the monsters has to be equal to the synchro
XYZ: Requires monsters of equal rank to the XYZ, the monsters are placed under the card instead of being destroyed, becoming "Materials". XYZ monsters have effects that let them discard a material to do something cool.
Pendulum: They're more a sub-type of card. They're a monster (Can be any of the above types, a normal or effect monster) but they can also be played as a spell in either your left or right spell zone, having their own effects when played this way.
When a pendulum monster is destroyed it's sent to the extra deck instead of the graveyard.
Additionally, pendulum monsters have "Scales" on them corresponding to a level. When you have two pendulum monsters in your left and right spell zones then you can special summon monsters whose levels are inbetween the scales. Pendulum monsters in the extra deck can only be resummoned this way.
Link:
Requires a number of monsters, often with a specific restriction, such as them being effect monsters of a specific level.
Link monsters have arrows pointing to other zones, which are called "link points" their effects target link points.
ADDITIONALLY: Now you can only summon monsters from your extra deck to a link zone or an extra monster zone, there are two of these zones in the center of the playing field, that are "Shared" by both players, however you can only use one one of them.
If a monster is brought from the graveyard it can go in a monster zone, but if it's coming from the extra deck it has to go in an extra zone or link point.