There wasn't any one person who invented baseball. Baseball came out of informal bat-and-ball games that had been played for centuries without any formal rules--more like "house rules" that were popular in whatever area you lived in. But these could change from game to game, depending on how many players you had and where you were playing. You might have four bases, or five bases, or two bases, or just one base, where the object was to hit the ball in the air and run to a tree. The first formalized game to come out of these casual games was cricket, with its rules first being written in 1744, though the modern game is more closely related to the 1788 rules written down by the Marylebone Cricket Club.

That game was imported into America and was quite popular in the United States during the first half of the 19th Century, but it wasn't the only bat-and-ball game played. This previous response goes through the reasons why cricket faltered, but suffice it to say that the early cricket clubs could be unwelcoming to non-English and non-British members, and the size of a field for a formal cricket match was not always available in urban settings, both factors which stymied the game's growth in the United States at a critical time, and both factors which baseball and baseball clubs were able to take advantage of.

In the first half of the 19th Century, there was a lot of migration from both overseas as well as rural areas into urban areas as industrialization took off and there were jobs to be had in the cities. Out of this, many athletic clubs were established in major cities to give young males something to do in their leisure time, and these at first weren't altogether different than an athletic club today. A member paid a fee which gave them access to the facilities and equipment, but importantly, there would often be pick-up games of various sports among the members. Though it should be said that not every club had actual facilities; some of the cheaper "clubs" just met on a field week to week, and you paid a smaller fee to join, which would be used to pay for equipment like bat and balls, and pay the owner of the field for access.

One of the more popular sports to be played by these athletic clubs was bat-and-ball games, but again, the style of play would vary depending on what field the club could get access to and how many players they had available for that particular game. A club that had formed by the early 1840s was the Knickerbocker Club, located in the midtown Manhattan. The club was made up of locals to the area, many working for the fire department, but it also included businessmen and lawyers and men from all walks of life. The president of the club was a guy named Duncan F. Curry, who worked in the insurance industry, and he described the early days of the club this way:

"For several years it had been our habit to casually assemble on a plot of ground that is now known as Twenty-seventh street and Fourth avenue, where the Harlem Railroad Depot afterward stood. We would take our bats and balls with us and play any sort of game. We had no name in particular for it. Sometimes we batted the ball to one another or sometimes we played one o'cat."

Duncan further explained that one of the other club members, Alexander Cartwright, came to the playing field one day with the idea of playing nine innings, with each half inning made up of three outs. The two teams would switch off between fielding and batting. Cartwright also suggested the diamond shape for how the bases should be laid out. Said Duncan:

"At that time, none of us had any experience in that style of play and as there were no rules for playing the game, we had to do the best we could under the circumstances, and aided by Cartwright's judgement."

Curry's assertion that Alexander Cartwright came up with the modern rules of 9 inning games and a diamond-shaped base path have since been disputed, notably in Baseball in the Garden of Eden: The Secret History of the Early Game by John Thorn. Thorn does concede, though, that Cartwright may have come up with the idea of 3-outs per half-inning.

Regardless, what happened next is undisputed: once the Knickerbocker Club had come up with a set of rules that they liked, they began playing regular inter-club games using the rules, eventually challenging other New York clubs using the rules (typically, they played two games, one game each with each club's preferred rules), and then, on September 23, 1845, they formally wrote them down and adopted them at a club meeting (although the earliest copy of these rules dates to 1848).

The early rules contained several differences from the modern game of baseball, though: the ball could only be thrown underhand to the batter ("pitched"), a batter was out if the ball was caught on the first bounce, and the game lasted only seven innings.

The Knickerbocker version of the game quickly spread in metro New York. In addition to the Knickerbocker club, other early New York area clubs playing baseball included the Eckford club, the Atlantic club, the Excelsior club (all in Brooklyn), the Gotham club (in Manhattan), and the Union Club (in the Bronx). Nevertheless, it was still a local game for a while longer.

In New England, they played a different form of baseball, which came to be called the "Massachusetts game", while in Philadelphia and the Midwest, they played a different variation called "town ball". They were all hit-a-ball-with-a-bat-and-run-to-a-base type of games, but they were each different in the layout of the field, in the manner of getting "out", and in the manner of scoring.

In 1858, an organization was formed in New York City to better help scheduled games between the growing list of local baseball clubs, which would also better help govern the games: formalize the rules, settle disputes, keep track of wins and losses, and also help spread the game outside of New York City. This organization was called the National Association of Base Ball Players, and out of this came formally published rules available all over country, available beginning in 1860 in Beadle's Dime Base-Ball Player. The early rules were based on the Knickerbocker Rules developed by the players of the Knickerbocker Club, but annual meetings of the NABBP began to revise them. And as the game spread outside of New York City, some non-New York rules found their way in the game that had been popularized first in the "Massachusetts Game": a ball had to be caught in the air to be counted as an "out", and the ball could be thrown overhand to the batter. Other rules were adapted throughout the years as well: nine innings per game, four balls for a walk (this had started at nine, then eight, then six, before being set at four in the 1880s).

The game's popularity took off after the Civil War, leading to an exit of some of the most successful clubs in the NABBP to form the National Association of Professional Base Ball Players in 1871, or the "National Association" for short. The amateur league didn't last long afterward, splitting apart into regional governing bodies, while this first attempt at a professional league didn't last much longer, folding in 1875. But some of the clubs survived and re-organized the following year as the National League, which remains at one of the two leagues that form Major League Baseball.

So while there were certainly individuals who played key roles, the true history of the "invention" of baseball lies not with an individual, but with a club called the Knickerbocker Base Ball Club in New York City in the 1840s, who formalized the rules that the modern game is based upon. But it wasn't only them, because the game went through some pretty drastic rules changes under the governing body of the National Association of Base Ball Players, before giving way to the National Association, and then the National League, where further revisions were made. It wasn't any one person, or even any one organization that "invented" the modern game. It was a series of organizations that each contributed different rules until the modern rule-set was established. By the end of the 1880s, the game would look familiar to a modern baseball fan, even though more rules would be changed later on (the Designated Hitter, for instance). But between the 1840s and 1880s, a succession of organizations contributed to the rules until the game developed, by and large, into the game that is still played today.

The most authoritative books on the early history of baseball are:

Baseball in the Garden of Eden: The Secret History of the Early Game by John Thorn

Baseball Before We Knew It: A Search for the Roots of the Game by John Block

Baseball's First Inning: A History of the National Pastime Through the Civil War by William J. Ryczek

All three books contain the same basic information as described above.

EDIT: I should add that there is a persistent myth that the game was invented by Abner Doubleday, but this is not true. All three of the aforementioned books debunk the myth, and this previous answer by /u/spartiecat gives the details. Further, Alexander Cartwright didn't invent it, either, but Cartwright at least was one of the people who helped shape some of the early rules of the game.