So, my understanding of this is basically as follows: the 1896 election was a realigning election that defined a lot about party politics in the United States today. Before 1896, the Democrats were basically the more conservative of the two parties, but William Jennings Bryan hijacked the Democratic nomination and pushed the party to the left of the Republicans. McKinley then won by outspending Bryan. At the end of the process, we had two conservative presidents in a row (Cleveland and McKinley) but the parties had switched sides. The way the election campaign was fought also had a big influence on contemporary politics.
How so? Also, how was Bryan able to simply take the the nomination from Cleveland, who was a sitting president? What were the differences and similarities between the two parties between then and now?
Prior to 1896, the political structure of the country basically appeared as follows: the South was solidly Democratic (despite a few Republican holdouts in Appalachia and a short-lived but serious third party effort from the Populist Party), the West was solidly Republican (despite a few Democratic holdouts in California and a much more serious third party effort from the Populists), and the North was competitive between the Dems and GOP, with New York usually being the swing state that decided the election. As a result, you'll regularly find elections in the 1870s and 1880s as being between popular Democratic politicians from New York running against Republican Civil War heroes.
What Bryan (a very young man in 1986) of Nebraska did that was so different was realigning the Democratic Party to basically absorb the Populist Party, to give up on challenging the GOP in the North and focus on populism to swing the West into the Democratic column. If you compare, say, 1888 to 1896 it's like night and day between the states Cleveland won that year and Bryan did in 1896 (barring, of course, the South for both). Needless to say, Bryan's message of encouraging inflation to wipe out debts and ease pressure on farmers did not play well with the banking interests in the Northeast and Bryan got clobbered in many states Cleveland handily won in 1892.
Bryan's cause (intentionally causing rampant inflation to wipe out debts by basing US currency on both gold and silver rather than just gold) was utter poison to the old elite of the Democratic Party that had supported Cleveland, and would greatly strengthen the GOP in the North for the next 20 years. Meanwhile, the strength of the Democrats in the West would wane in the same period, as more radical movements sucked up the oxygen on the left in the West (you might be surprised to discover some very vibrant socialist movements in states like Idaho, Colorado, and Oklahoma in the first two decades of the 20th century,).
EDIT: Just realized that you asked about how the campaign was conducted. Bryan wasn't the first presidential candidate to campaign for President (Stephen Douglas had in 1860), but Bryan turned it into a tradition. Bryan would go on train on whistlestop tours of the US to speak at large rallies while his opponent William McKinley ran a traditional front-porch campaign where he'd expect reporters to come to him to ask questions. McKinley won, of course, but the front-porch campaign style was clearly passe and would finally die in 1920 with Warren Harding's successful campaign.