What services do US embassies generally provide to citizens traveling abroad? In other words, what types of issues encountered while traveling can be taken to the embassy for assistance, and what areas will they generally not get involved with?
I am most familiar with the US Embassy in Delhi, India, since I was there for an extended period of time (several months). You can always visit the website for the US Embassy wherever you are traveling; they post a complete list of services for American citizens.
To summarize some of the big ones, US Embassies will help American citizens with the following:
They also provide an immense array of resources: the websites and physical embassy locations usually have information ranging from disaster preparedness / (historical) likelihood of natural disasters to lists of medical offices in their consular areas to international driving laws. I believe most of them have some sort of "tips for Americans living in [that location]" section, too, with other miscellaneous information.
US Embassies sometimes are a great cultural resource, too. (I can't say this applies everywhere, but it's true in India.) In Delhi there was the American Center, which has a library and hosts music and dance performances, lectures, literary readings, etc.
Additionally, it's a good idea to register with the local US Embassy when you are traveling - particularly if you're traveling to a somewhat dangerous area. If you register, you can provide them with a cell phone number where the Embassy will text you security and travel information. In the event of a terrorist attack or natural disaster, they have a record of whether you were in the country, where you were staying, and how to contact you. And your family / friends back home in the US can call the Embassy for help or information, too.
The US State Department has an FAQ about what it can and cannot do for American citizens abroad in emergency situations, particularly.
Those are the big things that I know of that appear to be true for most US Embassies, but I would always check out the website of whatever locale is relevant to you if you want specific, concrete information.