Anki is a spaced repetition flashcard program and it should be exactly what you are looking for.
In addition to just text it also supports images and sounds, so if you are studying a language you can add sound so you can hear how words are pronounced, or if you have a "Flag of Finland" question you could have a picture of the actual flag as answer.
... organizing larger sets of questions ("1000 words in language X")
Anki handles large sets of questions very nicely. My personal deck (Anki's term for sets of questions) that I've been building since I started using Anki in 2008 is up at over 11,000 cards and Anki is still very responsive and quick on both Android and Windows.
Automatic synchronization between devices: both cards and learning status
Anki supports synchronization between all supported platforms.
Android app. Either web or OS X version. Preferably iOS app
Anki is available for free (and as open source) on among others Windows, Linux, OS X, Android. There is also a web interface that can be used for any device that can browse the web. For iOS there is a paid version available, but if you don't want to pay for it you can still use the web interface on iOS.
Intelligent ordering and intervals for cards. Some apps out there just rotate all cards, while I have found that spaced repetition is a must.
Anki does spaced repetition.
Fast-to-use UI: skipping cards and choosing learning status must be fast and easy. For example requiring choosing actions from menu is not acceptable, but swiping/always visible buttons/tapping is fine.
The UI is very convenient with easy to press buttons on computers and phones and keyboard shortcuts for computers (press space to show the answer and 1, 2, 3 or 4 to indicate how well you knew the answer).
It's not possible to skip questions, but you can easily show the answer and quickly mark that you didn't know it or that you knew it very well depending on why you wanted to skip the question.
It is also possible to suspend cards if there are some cards that you don't want to see again, but don't want to delete.
Statistics for learning progress
Anki can give you graphical statistics on your past and upcoming reviews
Library of ready-made card sets. This really is nice-to-have, as creating cards is not the problematic part.
Anki have a large library of user made decks (card sets) so you should hopefully find what you are looking for there.
Reviewing on Windows. My deck is using a "three sided card" with the question on top, the answer below that and then finally the pronunciation at the bottom.
Statistics on Windows
Reviewing on Android. Here is a three sided card in the other direction with question on top, pronunciation in the middle and answer at the bottom.