How do global factors influence the economy in the US?

This question is a bit ironic, because the United States is almost certainly the one country in the world that is least influenced by the global economy. As the world's technological, military and economic hegemon (not to mention sovereign of the world reserve currency), the United States is one of a handful of countries that actually could afford to shut out other countries and ignore them if we chose to. Other countries depend upon us; but we do not depend upon them.

Actually we might be the only one; while China has as large a GDP as we do and a much larger population, they are more dependent on trade than we are. The European Union as a whole might be able to function closed off from the world, but no individual country within it could afford to do so. The US has one of the lowest ratios of trade to GDP of any country in the world (about 25%, compared to the world average of 60%), which makes sense in terms of the gravity equation because we have a huge GDP and are very far from most other countries.

That said, we are still significantly affected by global events.

The one most people know about is oil prices; because we are so dependent on imported oil, the international price of oil has a strong effect on the US economy in the short run. Actually this is less true than ever; I've linked a graph of our oil imports showing that they have been falling recently due to a combination of reduced consumption (more efficient cars, solar and wind power) and increased production (largely fracking). Yet we are still highly dependent on foreign oil, and have been for decades. Many economists believe that the US recession in the 1970s was caused by OPEC fixing international oil prices.

We also produce an enormous amount of exports and purchase an enormous amount of imports, and if people stop buying our exports or stop selling our imports we would have a problem until we adapted our domestic production accordingly. But again, one of the reasons we are so independent is that we actually could adapt our domestic production (with some loss in efficiency) in a way that a country like Nicaragua or Bangladesh cannot.

The US financial system is also tightly linked with many other financial systems around the world, such that a stock market crash in China or Russia can also cause a drop in markets in the US and slow growth in Europe can also create a drag on US industries.

This is the reason why it's so aggravating to economists when people talk about "shipping our jobs to China" or "staying competitive with Europe"; nations are not in competition with one another. When one nation prospers, others around it do as well. When one nation suffers, we all suffer. We're all in this together.