I haven't heard of these terms before for board games. What about normally known terms? What's "Euro Style"?
I avoided using normally known terms as most folks who don’t know board games won’t know the differences. To answer your question;
Euro Style
This genre started in Germany with anti-war sentiment. These focus on economic, infrastructure, building, or agriculture management. They don't rely on direct player interaction, and encourage skill over luck. Rarely involves elimination. This genre most often involves "The Long Con" mechanic/style, where you can do something with intentions/planning to profit more over the long term in lieu of short term gains. Players are almost never "out" in this genre unless they massively misplay, and its truly hard to tell for sure who's won/lost until the bell has rung and it's call counted.
Despite their focus on strategy and skill, they're often relatively easy to learn, hard to master games.
Coop Style
This genre is listed, and rarely have PvP involved (Some have recently started innovating with it as an option). It often involves players working together against the game itself, sometimes with Traitor/Betrayal mechanics and twists. Often, most or all players win or lose together, barring secret goals. Think of it like a MMO "PvE Raid", rather than "PvP Arena". The "AI", as it were, often involves randomization elements for both replayability and to ensure difficulty is a thing.
Hidden Traitor
This genre is partially listed, and it involves secret identities with secondary or ulterior goals to the main group, or the singular goal of killing/destroying/disrupting the main group. They often have some of the largest possible player counts, and usually the more the better it functions for balance and fun. While some can be long, a lot of them can fly comparatively quickly, if everyone involved knows how to play and what to do.
Worker Placement
The birthplace of "Meeple". In this genre, players have a pool of 'meeples' to represent their number of turns/actions available in a round. Player decision determines how they are used, and strategic planning and goal/resource prioritization is key. Next to Euro style, probably the LEAST reliant on luck for who wins/loses. The Big Brain 3000 IQ Players™️ will flourish with this genre. More often than not, there's limited resources/actions in a game and/or round, and players compete for them via meeple placements.
RPG
Yes, everyone thinks of D&D and Pathfinder for this, but its more than that. This genre, for board games, involves a lot of the cool parts of pen and papers without DMs or excessively long campaigns (sometimes). They're more structured than Pen & Papers, but much more approachable for newbies. Often heavily thematic in nature, and relying on leveling/typical RPG mechanics, they often have randomization elements like dice or cards to represent AI choices. Mice & Mystics is a classic multi-stage campaign with the party taking on roles as mice in a D&D-esque team comp with puzzles. Legacy of Dragonholt and Gloomhaven are other examples of this style.
Legacy
One of the newest genres, these are often more esoteric than most board games. They often eschew independent playthroughs for compound playthroughs where the new builds upon the prior. Your first playthrough might play as normal, with the outcome determining future bonuses' or penalties. Physical versions are (theoretically) limited use playthroughs, often due to 'destruction'/defacing cards and pieces as playthroughs build, but through the magic of digital versions, this can technically be avoided.
War Games & Wargaming
These are among your oldest styles. Wargaming relies more often on large maps to simulate expansive fields of battle to make you feel like a general peering from above for positioning and strategy. Each player has a faction that builds up their units to attack and destroy other factions. Its often highly reliant on strategy and planning, as well as dice rolls for battle.
Wargaming, however, is a different beast. From miniatures like Warhammer Fantasy & 40k to Battletech and “Clix” (Miniatures with dial wheels in them to represent stats), these are games that often have many years or decades of rules and re-rulings and revisions and new units and so on. You’ve likely heard about this genre if nothing else; they’re the games where singular miniatures can reach $1000-2000 or more, and often involve players having to build and/or paint their units from prepackaged parts or paints. This allows for more customization and attachment to your units, but the prices are often expensive comparatively, from $200-300 or more for a moderately built new army with no used purchases.
Technology Enhanced Games
Another new-ish entry, they've run the gambit from gimmicky (Monopoly's credit card millennial game) to somewhat more innovative. They typically involve smartphone apps nowadays, and obviously involve electronic usage in their play. Munchkin even has an optional app, but it is not "required". XCOM the board game's app is how your game is timed and alerts you to attacks, while DropMix uses phones or tablets on a smart reader board that plays and mixes music as you set down cards.
Roll & Move
Arguably one of (if not the) oldest genres, especially in 'modern' contexts of the last 100-150 years, this often encompasses most of our childhood classics. From Monopoly and Clue to Sorry!, Candyland, and LIFE! These often rely heavily on luck, and thus have fallen out of popularity. Talisman is probably one of the most recent of this style, and has done the most to encourage skill over luck as an option, although luck's still a factor. The skill involved is in manipulating the luck to be the most in your favor that you can achieve.
Deck-Builders
From Dominion & Mage Knight to Clank!, Star Realms, Resident Evil and beyond; this is up there as one of the more strategic styles of card/board game available today. They involve varying degrees of planning, strategy, and depending on game and play style, the option to rely on a bit of luck.
Generally most players start with either the same hand as each other, or the same starting hand per faction. From there, an often randomized variety of cards along with a few static core cards are in the “community pool”; every player takes turns playing the cards from their hands to increase either their buying power, wealth, score, health, or attack power. At the end of a players turn, they usually discards their hand, move what’s been played to their discard, and draw; upon reaching the end of their deck, they reshuffle their remaining discard pile into their new deck, and begin anew.
The goal of deck builders vary, from being coop or PvP to being even a hybrid mix of the two. They involve strategy, card counting, deduction (depending on the game), and balancing a fine line between a deck being too small or too big. Various strategies for tackling deck size/strength are available in most deck builders, from “milling decks” (Draw/Forced Enemy Discard focus) to “Summon decks” (Lots of things playing/summoning other cards into play) and more.
These games generally end upon either a score limit, round limit, player eliminations, or in some cases when certain card decks in the community pool are emptied out.
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