yugioh new mechanics overview

LINK MONSTERS:
Link monsters are extra deck monsters with effects that affect monsters in the zones pointed to by the red arrows going around the edge of the monster art. they have a link rating, which usually dictates how many arrows they have and how many tributes they need. They can't go into defense mode or be flipped face down.
so instead of being able to summon extra deck monsters (fusion/synchro/xyz/link) to main monster zones like you're used to, you now have 1 dedicated extra monster zone and can only have 1 extra deck monster at a time. however, if that monster is a link monster then you can summon more extra deck monsters to your main monster zones that have link monster arrows pointing to them.

PENDULUM MONSTERS:
Pendulum monsters are monsters that can also be played as continuous spell cards and have different effects for both ways of being played. They can be played in the leftmost and rightmost spell/trap zones. The numbers on the sides of them are their pendulum scales. When you have 2 pendulum monsters out, you can special summon monsters from your hand with levels between the pendulum scales. Say you have your pendulum zones filled, on one end a card with a scale of 4 and one end a card with a scale of 9. You can then special summon monsters from your hand with levels between 5 and 8. Whyen a pendulum card is destroyed it goes into the extra deck rather than the graveyard.

XYZ MONSTERS:
XYZ monsters are extra deck monsters that require you to tribute monsters of a specific level. Instead of having levels (the red circles with yellow stars) they have ranks (the black circles with yellow stars) so a rank 4 XYZ monster will require X amount of level 4 monsters to summon it. Instead of sending the tributed monsters to the graveyard, you overlay them on top of each other and stack the XYZ monsters on top. Most if not all XYZ monsters then have effects that typically require you to "detach" these overlaid cards and send them to the graveyard as a cost for activation.

SYNCHRO MONSTERS:
Synchro monsters are extra deck monsters that typically require some combination of a tuner monster (it'll say in the card type like "SPELLCASTER/TUNER/EFFECT" or something) and one or more non-tuner monsters. You have to use monsters whos levels add up to the total level of the synchro monster, so for a lv7 synchro you'd need a lv1 tuner and a lv6 non tuner, or a lv2 tuner and a lv5 non tuner, etc.

Sometimes the requirements for a specific synchro/xyz/link monster will be more specific than I outlined but you get the idea mostly