When you use Google Reader and you want to subscribe to a new feed, you will see this first:
Can I automatically get the information, that this feed has 179 readers? Does Google provide an API for this?
This post on StackOverflow describes the API Google Reader uses to get those subscriber details in its AJAX. It should be noted though that this is meant to be used internally by Google Reader rather than the general public, so is subject to change or removal without notice (much like the Google Weather "API" that disappeared a few days ago).
Feedburner also offers an API, however if it isn't your site then you are reliant on the site owner explicitly making their stats public. Plus the Feedburner API is being discontinued in about a month - without a replacement on offer at this stage.
If you own the site, you can get subscriber details through Feedburner or Google Web Master Tools. Plus every time the Google Reader bot crawls your feed, it lists the number of subscribers in its User Agent string which will appear in your server logs.