What were the crucial class differences in how people lived in Europe?: The upper class (aristocrats), the middle class (industrialists, bankers, shopkeepers, professionals), and the lower class (peasants, laborers, workers).
Why was it clear that socialism and revolutionary movements were popular among the lower classes? What differences were there in the various countries and why?
Why did so many people immigrate from Europe to Australia, the U.S., Canada, Argentina, etc. in this era?
While class differences have arguably always existed, the industrial revolution and urban development of the nineteenth century “made society less unified and more diverse” in Europe, widening the gap between the upper, middle, and lower classes. 
The upper class enjoyed little to no income taxes, were filthy rich (with the richest five percent of households in all of advanced Europe receiving 33 percent of the national income), and were generally born into their wealth, having time for leisure activities.
Interestingly, the middle class, which can be said to be perhaps the most diverse of the various levels of class, accounted for less than 20 percent of the European population. Different degrees of social status could be found within this middle class: the “upper middle,” “middle middle,” and “lower middle.”
The first level benefited from modern industry and progress, gaining income, household servants, and private coaches. Once embraced by the old nobility, these people “lost any trace of radical socialism [desired by the lower classes] as they were drawn to an aristocratic lifestyle.” 
The second level consisted of moderately successful industrialists, merchants, physicians, attorneys, and the like that lived comfortably but that lacked the “immense wealth” of the upper crust.
Finally, the lower level were white collar employees and small businessmen that made no more than their blue collar counterparts and owned no property, but that were “fiercely committed to preserving the distinction between themselves and lower classes.” 
The humblest class – the working class – was also partitioned into several sub-groups: the highly skilled, the semi-skilled, and the unskilled. However, collectively, these people comprised 80 percent of the population and received less income than the upper and middle class combined.
These individuals typically worked from sun up to sun down for meager wages to get by. They baptized their children, but did not regularly attend church. Instead, they drank, gambled, and played sports.
As evident by how drastically different (and difficult) life was for the lower classes, it is no surprise that socialism and revolutionary movements were popular among them. Nineteenth century liberals from these lower classes wanted more freedom – political, personal, and economic freedom. They wished for representative governments, demanded freedom of speech and the press, and believed in laissez-faire economics.
Beginning in Sicily, revolutions across Europe spread to engulf Italy, France, Germany, and the majority of the rest of the continent in the mid-1800s. Unfortunately, they were dismissed in due course and ended in repression.
In Italy (or what would become it), Italian patriot Guiseppe Garibaldi worked to unify the south while Count Camilo di Cavour, prime minister for King of Piedmont-Sardinia, Victor Emmanuel, drove out the Hapsburgs in the north. Though considered a “failure” at the time, Italian nationalists triumphed in that all of Italy was unified by 1870. Additionally Victor Emmanuel promised to work in tandem with an elected legislature and guaranteed the personal freedoms his liberals wanted.
In France, an insurrection followed a similar timeline as that of the earlier French Revolution of 1789 – from Bourbon monarch to a more limited monarchy to a republic to an empire.
The German states feasibly saw the most change – long-term unification under Prussia’s King Wilhem (William).
In all this turmoil of great social and political upheavals such as industrialization, overpopulation, and urbanization, Europeans began immigrating to places like Australia, the United States, and Canada in large numbers.
Persecution and economic hardships factored in as well. The 1840s and 50s saw mostly Irish, Germans, and Scandinavians leaving due to famine, religious persecution, and political conflicts. But the period beginning in the 1880s until the start of the first world war experienced the greatest departure of Europeans from their home countries.
In the United States alone, more than 20 million people entered the country from Southern and Eastern Europe;  the former searching for economic opportunities (the “American Dream”) and the latter (mostly Jews) for protection from religious persecution.
Throughout the course of the nineteenth century, it must be noted that the “Age of Progress” was well underway. Thus, even if disappointments were seen across the continent, the region was very much moving forward in some ways (such as in science and technology) and experienced some relative peace and prosperity among its peoples.
 Larry E. Gates, Jr., ed., “Rich, Poor, and Middle Class Life,” in Advanced Placement European History, www.historydoctor.net/Advanced%20Placement%20European%20History/Notes/rich_poor_and_middle_class_life.htm
 Bureau of the Census, Historical Statistics of the United States 1789-1945: A Supplement to the Statistical Abstract of the United States (Washington, D.C.: United States Government Printing Office, 1949), https://www2.census.gov/prod2/statcomp/documents/HistoricalStatisticsoftheUnitedStates1789-1945.pdf