This question originally appeared on Quora. 
Answer by Jason McDonald, Left bench as a player, but a lifelong fan of the game
Some of these overlap, it's not a typical A-B-C path between "eras."
Pre-1900 - is its own beast, because the basic rules of the game had not settled into place (examples: the number of balls for a walk was once as high as 8, foul balls did not count as strikes, so a batter could foul off pitches until he got one he liked, etc.) and the American and National Leagues had not established the clear supremacy as the two major leagues. Some of the individual teams existed prior to 1900, but 1901 is generally considered the start of "modern baseball."
The Dead Ball Era (1901 - 1920) -- This was baseball before power hitting became important. Lots of singles, bunting, stolen bases ... what we would usually consider "small-ball" tactics. Pitchers would routinely start 40 or 50 games, and win 25 or 30. A few things that contributed to it:
- The original rules basically played with a single ball as long as it was usable. The ball would wear down, become a little softer (hence, harder to hit well), and a little harder to see as it got dirty, which would favor the defense. The rules changed after the beaning death of Ray Chapman in 1920 so that new balls were used more often for better visibility. Cleaner, firmer baseballs helped the batters.
- The "spit ball" was outlawed, also in 1920. Pitchers in the early days were allowed to tamper with the ball -- spit, scratching up one part of the ball -- to give it odd aerodynamic properties that made it harder to hit. Individual pitchers who were throwing the spitball at the time of the rule change were grandfathered in, but otherwise the pitch was outlawed.
- New park construction. Several new baseball stadiums were opened over the course of the Dead Ball era, and some of them had hitter-friendlier dimensions. It could be that it took tactics a little while to catch up to the dimensions of the park.
World War 2 (1941 - 1945) -- Kind of its own little mini-era. A lot of the best players went into military service, leaving the teams behind with lesser players. Teams you wouldn't expect won titles (St. Louis Browns), and players you wouldn't expect won individual awards.
Segregation Era (1901 - 1947ish) -- Up until Jackie Robinson broke the color line in 1947, black players were not allowed to play in the major leagues. Robinson officially broke the color line in 1947, though the widespread integration took a few years beyond that.
Post-War Era/Yankees Era (1945 - late 50s/early 60s) -- The Yankees had an extended run of dominance after the war, through the 50s and into the 60s. From 1947 to 1962 (inclusive), the Yankees went to the World Series 13 times, winning 10 of them.
Westward Expansion (1953 - 1961) -- The Boston Braves moved to Milwaukee in 1953 to start the ball rolling, and then in 1958, the Brooklyn Dodgers and New York Giants moved to Los Angeles and San Francisco, respectively. The expansion California Angels, giving the American League its first west-coast team, could also be seen as part of this initial expansion west.
Dead Ball 2 (The Sixties, roughly) -- The balance of the game started swinging back toward pitching in the 60s, such that in 1968, Boston's Carl Yasztremski won a batting title with an average of .301 (the lowest in modern history) and St. Louis' Bob Gibson registered a 1.12 ERA. In 1969, the mound was lowered and the strike zone was reduced to bring offense back to the game.
Designated Hitter Era (1973 - current, AL only) -- In 1973, the American league adopted the designated hitter, which allowed a position player to hit for the pitcher. This has consistently added offense to AL stats (estimates put it around half a run per game).
Free Agency/Arbitration Era (1975 - current) -- Up until 1975, a team controlled a player's rights basically as long as they wanted him. He could be traded or released without any say by the player. The first (unsuccessful) challenge to this rule was the Curt Flood case, where the St. Louis outfielder tried to overturn a trade, but Flood's case (which ultimately went to the US Supreme Court) set the stage for 1975: Andy Messersmith (LA) and Dave McNally (Montreal) became baseball's first true free agents, stating they would play out the final years of their contracts and then sign with whichever team they wished. Subsequent negotiation between the players' union and MLB led to a three-tiered system -- young players were controlled by the team while in the minors, and their first few seasons, players between 3* and 6 years were eligible for salary arbitration, and players with more than six years' experience could declare themselves free agents.
*=the rule was later adopted to add one-sixth of 2nd-year players with the most major-league service time to the arbitration list. This is why you'll sometimes see a team keep a promising prospect in the minors until June -- it buys the team an extra year of pre-arbitration.
Steroid Era (unknown, but late 80s - 2005 seems likely) -- obviously we can't know with certainty who was taking what, but it seems around the late 80s that offense, and power numbers in particular, took a dramatic upswing with no corresponding rule change that would account for it. A LOT more 40 and 50 homer guys at first, followed by the increasingly frequent assaults on Maris' record. In 2001, some testing was started in the minors, and survey testing (testing with no punishment to gather data on the scope of the problem) in the majors began in 2003 but MLB did not negotiate suspensions for violations into the player agreement until 2005, so that's probably the best place to mark the "official" end.
Wild Card Era (1994 - current) -- Until 1994, you had to win your division to play in the MLB playoffs. This led to several frustrating situations where a second-place team in one division had a better overall record than the division winner in another division. In 1994, MLB took a page from football's playbook and added a wild card team -- i.e. three division winners, plus the team with the best overall record regardless of division. They've since added a second wild-card, but the wild-card has definitely changed how teams approach the season.