A Region is a physical area, a geography, in which the adventure takes place. A generation refers to the new Pokemon, Moves, Abilities, and other mechanics that are implemented in the game.
Most games are contained to a single region. For example, Ruby, Sapphire, Emerald, Alpha Sapphire, and Omega Ruby all take place in the Hoenn region, and only the Hoenn region. The same is true for Red, Blue, Yellow, Fire Red, and Leaf Green. They all take place in Kanto region, though Fire Red and Leaf Green also added what could be called a mini-region, the Sevii Islands.
Gold, Silver, Crystal, Heart Gold, and Soul Silver, however, are a little different. They all start in the Johto region, but also include the Kanto region. There is a point in the game where you unlock the ability to travel back and forth between them freely.
Another important thing to consider is that different generations are only ever forward compatible, never backwards. That is to say, through Pokebank and similar out of game systems, you can move Pokemon forward from an older generation to a newer one, but you cannot move them from a newer one back to an older one. Some of this is due to software limitations or hack/glitch abuse prevention, but a fair amount of it is also balance changes. Things like changing a Pokemon's Hidden Ability (like Chandelure from Gen 5 to Gen 6), or alterations to a Pokemon's stats, even removing a selection of moves, both from specific Pokemon or from the game as a whole (Sword/Shield did this pretty extensively).