If we use a website (Wikipedia, for example), do we enter into a contract or a promise with the site owners (that we will fully abide by the terms and conditions of the site)? And does the failure to do so warrant an expiation as one has broken a promise? Please clearify, as I am in doubt whether not following the rules of the site is considered breaking a promise or not given that breaking a promise is a major sin.

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

The users should abide by the terms and conditions of the website as long as they are not against the Islamic sharee'ah. Allaah, The Exalted, says (what means): {O you who have believed, fulfill (all) contracts....} [Quran 5:1] The Prophet, sallallaahu ʻalayhi wa sallam, said, “Muslims are bound by their conditions, except the conditions that make what is lawful unlawful, or what is unlawful lawful.” [At-Tirmithi classified it as hasan saheeh (good or sound)]

As for the expiation for breaking an oath, there is no expiation due in this case nor is it legislated to begin with when the user violates the terms and regulations of using websites and the like.

Allaah knows best.