Hi Alex,
This process is typically referred to as "reverse geocoding" where you get a location from the user as a latitude and longitude point (via their smartphone or web browser) and then use a "reverse geocoding" API to convert that to an address and/or a list of administrative divisions. If you were to start with a human readable address and then want to look up all the administrative divisions for that address, you would might need to first use an API to "geocode" that address to convert it into a latitude and longitude and then use that lat/long to do the "reverse geocode" operation - although many APIs allow you to do those two steps with one API call.
My understanding of the situation is the following:
1. there aren't many great government sources for this domestically nor worldwide, but there are some usable ones worth knowing about
2. there are some decent free non-governmental sources (which use data derived from government sources)
3. there ought to be better sources from the government and there are some emerging opportunities that look promising
U.S. Government Sources
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The best US Gov't API for domestic administrative divisions is probably the Census Geocoder, but it's not great because it relies on Census surveys which are not updated nearly as often as many non-governmental sources (even though many non-governmental sources incorporate Census data and even use it as a baseline)
The Census Geocoder will take human readable addresses if they're relatively well formatted. It will also take normal (Web Mercator WGS84 SRID 4326) lat/long points. Here's an example of a search with a lat long in San Francisco
The best direct source from the US Gov't for international locations (that I know of) is GEOnet Names Server (GNS) but I don't think there's a simple REST API for it and I also don't think the associated location data covers administrative boundaries as polygons, I think it's just names and points on a map. In any case, here's more info:
http://geonames.nga.mil/gns/html/index.html
Non-Governmental Sources
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GNS does however help provide the basis for non-governmental aggregators like GeoNames.org which do allow you to get the different administrative boundaries that a point falls within. Here's an example of a GeoNames.org API query for a lat/long point in Ho Chi Minh City, Vietnam:
However at the international scale it currently looks like OpenStreetMap has done the best job of coalescing this open data for administrative divisions and making it easily available as an API. OpenStreetMap provides this query from their Nomatim API. Here are the examples for the same lat/long points
The same OpenStreetMap data is also incorporated into a number of other APIs. MapQuest also provides a great free API using the same data and depending on what you need it may be provided and augmented in a better way than Nomatim. After you create an account and get an API key from MapQuest (needed within URLs like the one below) you can compare the results for Vietnam: