Well, there's Steel, Dark, & Fairy types now. Certain super-effective match ups have changed. Some Pokemon can Mega-Evolve when holding the correct stone. Now there are these Z-moves.

You can find all of the specifics on these topics using a Google search.

What's probably most important to realize is a Pokemon's move does 1.5x more damage if the move is the same type as the Pokemon. Even a higher multiplier is applied if the opponent is weak against that type.

However, each Pokemon move is also classified as either a physical or special move in addition to its type. Physical moves use Attack and Defense stats to apply damage, while special moves use Special Attack and Special Defense to apply damage. You will have trouble using a Pokemon with low Special Attack and Special Defense stats if you only teach it Special moves and vice versa.

Finally, an important change came to status conditions. Certain types can't be afflicted by redundant conditions. Electric types can't be paralyzed, Fire types can't be burned, Ice types can't be frozen, Poison types can't be poisoned, Grass types can't be affected by powder moves like Sleep Powder, and Ghost types can't be forcibly removed or kept from battle from moves that would do so (I think).