Search your library for a green creature card with converted mana cost X or less, put it onto the battlefield, then shuffle your library. Shuffle Green Sun's Zenith into its owner's library.
If Green Sun’s Zenith is countered, none of its effects will happen. Notably, it will be put into its owner’s graveyard rather than shuffled into its owner’s library.
In most cases, if you own Green Sun’s Zenith and cast it, you’ll shuffle your library twice. In practice, shuffling once is sufficient, but effects that care about you shuffling your library (like Psychogenic Probe, for example) will see that you’ve shuffled twice.
If you own Green Sun’s Zenith, but an opponent casts it (due to Knowledge Pool’s effect, for example), that opponent searches their library for an appropriate creature card, then shuffles that library. That opponent then shuffles Green Sun’s Zenith into your library. You won’t shuffle any library in this case.
If this spell doesn’t resolve, none of its effects occur. In particular, it will go to the graveyard rather than to its owner’s library.