This one could be interpreted a number of ways, and most of the interpretations could be answered with a book. I'm going to take a shot at it anyway.
In Marxism, the bourgeoisie is the class that owns the means of production. This can be direct ownership of, say, a factory or train; or it can be a nice stock portfolio. The key point is that you are deriving most of your income, not from your own labour, but from charging others for the use of your property. The proletariat is the class of people who earn their money by working.
'Middle class', on the other hand, could be taken to refer to white collar workers - people who do mental as opposed to physical (blue collar) work.
There is a parallel with the bourgeoisie, which I don't think Marx foresaw (I'm not all that familiar with his work though) - white collar work tends to require a certain amount of education, which can be viewed as a one time investment which the 'worker' then uses to generate income for the rest of his life.
I think those are reasonable definitions for this question, because the decades around 1900 did see a pretty big shift in power away from the large scale factory owners, and a simultaneous growth in the number and importance of white collar workers. Actually, wikipedia says that the term 'white collar' was coined in 1911, so there you go.
Again, this is a huge issue, but to touch on a few interesting points:
- the proliferation of unions, child labour laws, worker health and safety laws, and so on was both a cause and a reflection of the factory owners' decreasing power.
- the increasing number of people interested in improving themselves mentally led to libraries and museums being built, or opened to the public.
- the first MBA program (Harvard's) was offered in 1910 - a hundred years previously it would have been unthinkable for a first rate university to provide vocational training for businessmen.
- public education became compulsory in Britain and most of the United States during this period. There was a lot of experimentation in both the content (ie, curriculum) and methods (teaching techniques, for example Dewey's progressive theories in America) of education.
- women gained economic, and therefore political, power as well-paying nonphysical work became more available - this led not only to their getting the vote, but also to things like prison reform and Prohibition, which tended to get a lot of female support.
- the growth in office jobs also made it much more difficult to get servants at an affordable rate, which in turn led to a fashion for simplicity - an upper class or bourgeois Victorian woman didn't have to dust her huge collection of knick-knacks, or assemble her massively complicated outfits and hairstyles, while factory workers and servants didn't try to be fashionable. Fashions for middle class people couldn't require an army of servants.
- and here's one I never heard of before: apparently restaurant practices changed significantly to reflect the preferences of their new clientele.
Ok, this is a ridiculously long post, and it only skims a lot of complicated subjects. I hope some of it was interesting for you - feel free to ask more questions or narrow this one down if this wasn't what you wanted to see.