I’ve played regularly, mildly competitively, for the last 10 years.
An overview of the blocks since Tempest block:
Urza’s (Saga, Legacy, Destiny): The block is full of broken tutors and overpowered engine cards. Decks abusing Tolarian Academy, Tinker, Replenish, Yawgmoth’s Bargain and other cards dominate many constructed formats.
Masques (Mercadian Masques, Nemesis, Prophecy): Wizards powers down this set considerably compared to previous sets, and most players dislike these sets pretty strongly. Rebels (creatures with activated abilities to search for more of their kind) make their first appearance.
Invasion (Invasion, Planeshift Apocalypse): The first of the great Multicolor blocks. All three sets contain lots of multicolor cards, and encourage people to play lots of colors. The big deck at this time is based around Fires of Yavimaya and big creatures.
Odyssey (Odyssey, Torment, Judgement): This block is based around using the graveyard as more of a resource, with cards that can be played from the graveyard, and cards that care how many cards are in your graveyard. The other interesting bit is that it contains the first sets that are not color balanced. Torment has more black cards than any other color, and Judgement has more Green and White cards.
Onslaught (Onslaught, Legions, Scourge): Introduced morph. Introduced storm, which, as an above poster mentioned, has completely redefined what it means to be a combo deck in most cases. Legions was an all-creature set. This was the beginning of the modern creature type push, where creatures have multiple types (race, class), and sets generally focus on a handful of creature types that interact together.
Mirrodin (Mirrodin, Darksteel, Fifth Dawn): The artifact block. Introduced “indestructible” cards, and a deck that’s been a tournament powerhouse in most formats since: Affinity. Fueled by artifact lands and other cheap artifacts, this deck can easily drop it’s entire hand by turn 3, and sometimes just win by then, too. Introduced equpment, which are artifacts that creatures can wear like enchantments, but unlike enchantments, can be moved around and stay in play if the creature doesn’t.
Kamigawa (Champions, Betrayers, Saviors): Lots of legendary cards, and a new legend rule to go with them (if multiples of the same legend are in play, they all go to the grave). The most important thing to come out of this block, is Umezawa’s Jitte, a truly absurdly good equipment.
Ravnica (Ravnica, Guildpact, Dissention): The ten two-color pairs represent Guilds in this block, with 4 appearing in the first set and 3 each in the others. Each guild has a characteristic flavor and ability word. Introduced hybrid mana, which is a mana symbol that can be paid by either of one or two colors, allowing “multicolor” cards that are either/or, rather than both. Introduced Dredge, which has become a powerful archtype in every constructed format.
Time Spiral (Time Spiral, Planar Chaos, Future Sight): Nostalgia block. The theme was past-present-future. The first set had a “reprint” rarity, in which one of 121 older cards was reprinted. The second had “color-shifted” cards, where older cards were reprinted, except with a different color. Serra Angel in blue, Ball Lightning in green, and Giant Growth in red, for example. The third set had “Future shifted” cards, which are reprints from future sets. Some of those cards have since been printed, some will be in the future, some probably never will be. The big standout from this block is Tarmogoyf, an undercosted green creature that led to almost every deck in Legacy splashing green.
Lorwyn (Lorwyn, Morningtide // Shadowmoor, Eventide): The four-set block. Actually, more like two two-set blocks. Lorwyn/Morningtide was heavily tribal, focused on creature race and class. Shadowmoor/Eventide was focused on color, with lots of cards that care if you have other cards of certain colors. Shadowmoor, Eventide was a heavily hybrid block, with the first having allied colors, and the second having enemy colors. Introduced the first new card type in a while, Planeswalkers, which are permanents with powerful abilities that can be directly attacked by creatures.
Alara (Shards, Conflux, Reborn): There are 5 3-color shards that start out separately, but end up together in the first all-multi-color set (Reborn). It’s early to say what the long-term impact of this set will be.
Some more general notes and trends:
In 1997, there were three main constructed tournament formats: Type 2, Type 1, and Extended (sometimes called type 1.x). To this was added Legacy (Type 1.5), which was Type 1 with all the restricted cards banned. Later, Legacy would have it’s own banned list that was distinct from Type 1’s banned and restricted lists. Extended began a rotation policy like Type 2, going through a few iterations, and now is “all the cards from the last 7 years”.
When 6th edition came out, the rules were overhauled, resulting in a much more consistent and coherent ruleset. Major changes: tapped creatures deal combat damage, tapped artifacts still work, interrupts are all instants, players lose immediately if they hit 0 life, and spells and abilities are processed on a stack, rather than playing in batches.
Some common abilities have shifted from color to color. Fast mana (rituals) have shifted from black to red. Temporary control effects and spell retargetting/duplication have shifted from blue to red. “Good cards” are still pretty solidly in blue.
Overall, the trend has been to increase the power level of creatures and creature-based strategies, while decreasing the power level of combos and non-interactive strategies. Older strategies of “counter almost everything”, “lock the game down”, and “combo out” are much less represented in Type 2, although they’re still present in older formats. For the most part, they’ve succeeded, with creatures from the past few blocks making splashes all the way back in Type 1. Most constructed matches are decided by creatures actually attacking, and even blocking! Many modern creatures have comes-into-play abilities that make them play like a spell with a body attached. Only the most efficient creatures without special abilities see any play at all.