Erwin Chemerinsky (my boss) in The Atlantic:

Campaign finance. In recent years, the Roberts Court has struck down a number of federal and state laws regulating campaign finance. Most famously, in Citizens United v. Federal Election Commission in 2010, the Court held that corporations have the right to spend unlimited amounts of money to get candidates for public office elected or defeated. It is likely that a Democratic majority on the Court would overrule this highly controversial decision. Just seven years earlier, in McConnell v. Federal Election Commission, the Court had upheld the very provisions that were declared unconstitutional in Citizens United. A Court controlled by Democratic appointees would likely overrule Citizens United and say that it is returning to its earlier approach.
Congressional power. Conservatives on the Court long have wanted to limit the scope of congressional power. For example, in 2013, in Shelby County, Alabama v. Holder, the Court struck down key provisions of the Voting Rights Act of 1965. This was the first time since the 19th century that the Court invalidated a federal civil-rights law dealing with race. In many states, it has led to significant new obstacles on the ability of minority voters to participate in elections. A Supreme Court with five Democrats would likely reverse this ruling and also be much more likely to uphold congressional power to regulate interstate commerce and to tax and spend for the general welfare.