Lively ball era
The first "lively ball era" started in 1893, when the pitching distance was increased. It was especially high in 1894, when National League teams scored a league record 7.39 runs per game and the league batting average was .309. 5 players that season batted above .400, including a record .440 by Hugh Duffy, and Billy Hamilton scored an astounding 192 runs. The league ERA that year was 5.32. Scoring slid for the rest of the 1890's but was still high by historical standards as late as 1900.
The 1920's and 1930's were also a lively ball era, so much so that an unqualified mention of "lively ball era" probably means this time. Scoring reached a peak in 1930, when National League team Runs/Game was 5.68 and the league batting average was .303 and the league ERA was 4.97. The scoring in 1930 was seen as excessive, and the official rules were changed to try to bring offensive levels down. A notable change was that balls bouncing over the fence were no longer counted as home runs; they became ground rule doubles instead. The National League was more effective in bringing scoring down than the American League. Between 1931 and 1939, NL scoring never topped 5.00 runs per game, while AL scoring never fell below that mark.
The era around the year 2000 may be seen eventually by baseball historians as another lively ball era. In 2000, the American League team Runs/Game was 5.30 and the league batting average was .276. The league ERA was 4.91.