An exhibition game is the North American term for a sporting event in which there is no gain or loss from whether the competitors are victorious or not in the competition. The comparable British term is friendly match.
In North American sports, exhibition games usually take the form of a handful of preseason games that are intended to familiarize teammates and get them ready for upcoming matches. In professional sports, preseason games also help teams decide which players to keep for the regular season. Occasionally - though more commonly in other countries - exhibition games are parts of larger events; particularly fundraising for charity, or as over-the-top demonstrations of skill for the purposes of entertainment.
Major League Baseball's preseason is known as spring training. All MLB teams maintain a spring-training base in Arizona or Florida. The teams in Arizona make up the Cactus League, while the teams in Florida play in the Grapefruit League. Each team plays about 30 preseason games against other MLB teams. They may also play exhibitions against a local college team or a minor-league team from their farm system.
Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington, D.C. for the 2005. It also used to be commonplace to have a team play an exhibition against Minor League affiliates during the regular season, but worries of injuries to players, along with travel issues, have made this very rare. The annual MLB All-Star Game is played in July between players from AL teams and players from NL teams.
More info is needed on the Japan All*Star Series.