To give you the quick run-down, Link cards drastically changed how Extra Deck Monsters (Fusion, Synchro, XYZ, and now Link) are summoned. They can now only occupy an entirely new space called the "Extra Monster Zone" which by default are two spaces in the middle of the field that are both shared by you and your opponent. In order to expand the number of Extra Monster Zones on the field, you must perform a Link Summon and place a Link Monster to any pre-existing Extra Monster Zones. Link Monsters have arrows pointing to which Monster Zones can now be used as Extra Monster Zones. Note that Link Monsters placed in the Link Zones may also give your opponent access to an Extra Monster Zone depending on their Link Arrows.
For Pendulum Monsters, as you are aware they can either act as a Monster or Spell card each with their own respective effect, thus both rules for play are applied normally depending on how you want to use the card. When used as a Spell however, they can only be placed in the Pendulum Zones which occupy the far-most left and right Spell Zones. When two Pendulum cards have been placed in each respective Pendulum Zone the player is now able to perform a Pendulum Summon, allowing the Special Summon of however many monsters whose Levels are between the two numbers of the Pendulum Scale (the numbers that the Pendulum Effect is written between). These monsters can be summoned either from the hand or from the Face-Up Extra Deck (when Pendulum Cards are destroyed, they are sent face-up to the Extra Deck).
This may all sound complicated, but this is coming from a person who has yet to play the Link format. Although I am interested, I haven't had the time to build a deck and try it out.