There are, of course, many different aspects to the First Amendment and there have been many cases that have been important with respect to each of those aspects. For this question, let me give you two cases that have had impacts on two very different areas of American life.
The first of these is Citizens United v. F. E. C., which was decided in 2010. This was the case in which the Supreme Court said that it was illegal for Congress to limit how much money corporations could spend on political campaigns. This has had a major impact on our political system which, in turn, impacts our lives. This ruling made it so that “Super PACs” could be set up to put unlimited amounts of money into campaigns for or against various candidates and issues. The long-term effects of this decision are not yet clear, but it has clearly changed American political campaigning.
The second case is Miller v. California decided in 1973. In this case, the Supreme Court ruled on what materials (books, movies, etc.) can be considered to be obscene and therefore not protected by the First Amendment. The decision said, among other things, that a work can only be considered obscene if it A) "depict(s) or describe(s) sexual conduct in a patently offensive way" and it B) "taken as a whole, lack serious value and must appeal to a prurient interest in sex." The effect of this decision has been to ensure that essentially all pornography that does not involve children is legal. This has had a huge impact on the internet and on the lives of any parents who have children who use the internet.