The Regular Season
The Major League Baseball season runs from the start of April to the end of September, with each club playing 162 games. That means about one day off every ten days, so baseball is pretty much "game-a-day".
Teams normally play "series" of three (occasionally four) games on consecutive days against the same opponent, normally with a "homestand" of two or three series, or a "road trip" (though most of the travelling is now done by air!) of two or three series. So a fan can expect to have a week of home games, and then nothing but Baseball on TV for the next week or so.
If a game is "rained out", then it's replayed later in the season (unless at least five innings have been completed for both teams, in which case the score is "called" at the point at which both teams had completed the last innings), usually as part of a "double header" (two games played on the same day).
Nowadays a double header means one game in the afternoon and one in the evening (the "night cap"), which is particularly tough on a pitching staff (as there's no time for rest at all). In the past double-headers used to sometimes be scheduled (for example on public holidays) with fans getting two games for the price of one. In modern day Baseball, however, they sell separate tickets for each game and generally there is at least a couple of hours between games, not least to allow the first game's crowd to leave and the second game's crowd to come in.
In modern day Baseball most midweek games are typically played in the evening, under floodlights (to allow people to come to watch the game after work), and often weekend games are played during the afternoon. However, if television wants a game to start at a particular time, then generally it'll start when television dictates!
American and National League
In Major League Baseball there are two "Major Leagues" - the American League (AL) and the National League (NL), each of which is split into three divisions. Almost all of the games played by teams are within their own league (half within their division, half against teams in the other two divisions), although since the late 1990's a small number of "interleague" games are played.
The main difference between the American League and the National League is the Designated Hitter rule (in use in the American League, so pitchers do not hit for themselves, and not used in the National League). When an AL team meet an NL, the Designated Hitter rule is used for games played in AL ballparks, and not for games played in NL ballparks.
Each team's first aim is to win their divisional title, and if they cannot win their division, then to finish as the best runner-up in their league (the Wild Card). If two teams are tied for a divisional title, or the wild card spot, then a one game playoff is held (venue decided by tossing a coin) the day after the season ends to determine the winner (potential coin tosses are held a few days in advance to allow the teams to make contigency plans).
The Farm System
Every Major League team has what is known as a "farm system", a set of lower-league "affiliates" playing in "Minor Leagues" whose job is to provide replacement players, and groom and develop up-and-coming talent into Major League players. Injured players often play for the Minor League team a few games whilst they "rehab" to get to full fitness.
There are generally three standards of Minor League baseball, "Triple A" ("AAA") - the players closest to the Major Leagues, "Double A" ("AA") and "Single A" ("A") for the real newcomers to professional baseball, just learning their trade. It can take a player anything from two to fifteen years to progress from the Minor Leagues to the Majors, and many players never make it at all. Most teams generally have one "AAA" team and one "AA", but it's quite common to have more than one "A" affiliate.
The Minor League teams are spread all across the country, often nowhere near the Major League club, and generally are located in smaller towns and cities where they can draw support of their own. Minor League Baseball isn't glamourous, and the results are rarely reported. The key for any player is that it's his step to the "Big Show" - Major League Baseball. Players aren't paid much in the Minor Leagues, and if they want to make money, they have to make it to the big league ball club.
When a player is replaced on the Major League Team's roster, he is normally sent back down to the Minor Leagues (though there are complicated rules which determine how many times this can done, and after a certain limit, a player has to "clear waivers" where any other clubs have the option to pick him up when he is sent back down to the Minors), and if he's injured he'll be placed on the "Disabled List" (DL).
Minor League teams do occasionally swap their affiliations from one club to another, and some Minor League teams aren't affiliated to any Major League ballclub - these are known as "Independents".
Almost all players are initially signed to clubs from the draft, in which clubs select eligible players in reverse order of their previous season's standings (so the worst team gets first pick, etc). However, in Baseball it takes so long for players to develop and reach the Major Leagues, that a team's draft position is rarely important - it's far more important to have a good scouting network and sign "good prospects". The draft is held in June, often almost un-noticed.
A much more immediate way to improve the Major League team is via Free Agency. When a player has been playing Major League baseball for a number of seasons (five usually) and reaches the end of his contract, he is able to "file for free agency" and effectively sign for whichever team offers him the most money (assuming that's his priority, and it usually is!). The winter "off season" free agent signing period is often the most important time for determining a club's potential for success the following season.
Free agency has only existed since the mid 1970's - before then a "reserve clause" existed which allowed clubs sole rights to a player, preventing him negotiating with other clubs. Since the abolition of the reserve clause player salaries have leapt, and the current restriction on free-agency is only by agreement between team owners and the Players' Union..
The most common way a team can improve its roster during the season is via a trade, in which the rights to one (or more) players are traded to another team for the rights to one (or more) of their players (and sometimes for cash). This becomes particularly frantic as the "Trading Deadline" approaches at the end of July.
A very common occurrence at the Trading Deadline is for a struggling team to "trade for prospects". With two thirds of the season gone, if a team isn't in the playoff race, then it may trade one or more of its "star players" (who will probably be ageing, possibly heading for free agency, and possibly overpaid) to a "contender" in exchange for "prospects" (younger players, often at AA or AAA level) who won't be useful now, but may be in a year or two. They'll probably be cheaper as well - "salary dumping" is often a priority for a struggling team!
The Trading Deadline is one of the key times of the season. A contender has to decide whether to mortgage some of the future to add one or two key players now, whilst a struggling team get the chance to accelerate their building for the future. For a team on the margins of a playoff race they have big decisions to make - do they go for it or play safe?
A veteran with more than ten years experience in the Major Leagues will normally have a "no-trade" clause in his contract, allowing him to veto such trades, though quite commonly he'll "waive" it (given the trade is probably moving him to a more successful club with a chance of playing in the post-season).