Well, most of the general trends in this country will probably continue. None of which have anything to do with Obama.
The economy will probably pick up -- we're still in a boom and bust cycle -- but middle-class manufacturing jobs will continue to be outsourced to other countries and machines. We keep having these recoveries, but they leave more and more people out. Obama may be able to take away the tax break for companies that send jobs overseas (which Republicans have assiduously defended), but that's not likely to stem the tide.
I'm sure there'll be more instances of extreme weather, like Hurricane Sandy. And unbearably hot summers. Climate change is real, it's happening, and it's getting worse and worse. Obama may be able to get the nation to participate in international cap-and-trade legislation (I sure hope so, but I'm not holding my breath), but that won't change things right away, and it may not be enough to do a thing. In the case of individual disasters, FEMA will at least be functional.
If Obama is able to rescind the Bush-era tax cuts on the super-wealthy, then the national debt will level off. If not, it won't.
I would love to see increased regulation on investment banks (like we had before the repeal of Glass-Stiegel [sp?]) and some bankruptcy/foreclosure reform. Not sure if we'll get it.
The US may participate in some multilateral military interventions (like in Libya and Bosnia) that are highly effective but don't get the nation's testosterone going. We're less likely to have a unilateral "war of choice" (like Iraq) that gets people crazy with excitement but turns out to be a disaster.
Ruth Bader Ginsberg will probably retire from the Supreme Court and be replaced with someone with similar political leanings. The Republican appointees are pretty healthy, so I don't foresee a net change. Obama will make some lower-court appointments that may bring about a subtle judicial shift, but nothing dramatic. In other words, Roe v. Wade will not be repealed.
The extraordinary Civil Rights movement of LGBTQ Americans will continue. People will begin to be embarrassed about their past views and pretend they always supported gay marriage.
Congress will remain bitterly divided and nonfunctional. Americans will hate Congress and reelect their own congesspeople.
Each year will bring a new high-tech toy that everyone needs right away.
Every few months there will be a shocking scandal of some sort. People will talk about it with great excitement and then they won't.
In 4 years, there will be another presidential election. Half the country will be outraged and bewildered by the other half, and vice versa.
Puppies and kittens will still be cute.
In other words, ditto Suz and Dawn. I could pretty much answer any question on this site by saying ditto Suz and Dawn, among others.