Hmm...where to start?
Well, to put it simply, the game has evolved into a need for card advantage. Take a look at any card in a recent set, and it will most likely involve Special Summoning, drawing cards, or adding stuff from your deck/graveyard to your hand. Archetypes of these days are usually just variants on how to gain the most advantage.
Konami has released 3 new monster classifications since 2003: Gemini, Synchro, and Xyz, and 2 new monster types: Divine-Beast and Psychic.
The first classification, Gemini monsters, go in the main deck, and are treated as normal monsters until you use up your turn's normal summon to summon them a second time to gain their effects. They can be difficult to use, but most Gemini monsters have very powerful effects that compliment their difficult cost.
The second classification goes in the Fusion Deck, now renamed the Extra Deck. Synchro Monsters require you to send 1 (or more, in rare cases) Tuner monster you control to the Graveyard, and 1 (or more) other monsters you control to the Graveyard, and the levels of the sent monsters must match the level of the monster you're trying to summon. Synchro Monsters redefined the game as a game of speed, and almost 200 have been released to this day.
The third classification also goes in the Extra Deck. Xyz Monsters are similar to Synchros in the sense that you require monsters of certain levels, but all you need is 2 (to 5, depending on the card in question) monsters of the same level (and sometimes need to share a type, attribute, or an archetype). Instead of sending them to the Graveyard, they "Overlay" into 1 monster card zone, where the Xyz Monster is placed on top of. The "Overlayed" monsters stick around as Xyz Materials, which the Xyz Monsters typically use to activate their effects. Xyz Monsters additionally have no level; they have ranks instead. Though a cool concept at first, they quickly redefined the game as "Who can get out this 2400 ATK Solemn Judgment with 1 card before their opponent can?". Many Xyz Monsters are still very fun to build decks around due to their generic requirements, and with over 230 existing in the game so far (with more to come), you have many options to choose from.
Divine-Beast type monsters are just for the Egyptian God Cards--yep, they're legally playable and not just different-colored cards with worthless flavor text.
Psychic type monsters are unique in the sense that they're almost an archetype in themselves. They focus on paying life points or banishing (removing from play) themselves for more card advantage later in the duel.
