When the precedence between names or nomenclatural acts cannot be objectively determined, the precedence is fixed by the action of the first author citing in a pulished work those names or acts and selecting from them; this author is termed the "First Reviser". (24.2.1)
This is an extension of the Principle of Priority as I discussed earlier, and it's meant to clear up controversy over the precedence of names when either the publication date, original spelling or publication of synonyms in the same manuscript makes it unclear which name has priority and is therefore valid. The First Reviser is the first person to cite the work(s) in question in a published manuscript and state which name is valid or available.
Linnaeus's Systema Naturae (1758) has many examples of this principle, since so many of the names were later found to be subjective synonyms (i.e. they referred to the same species as determined by later authors). One would be the snowy owl, which was given two names (Strix scandiaca and S. nyctea), later determined to be synonyms. The First Reviser, Lonnberg, gave precedence to S. scandiaca, which became the fixed name upon his publication in 1931 (Note: it is not the /valid/ name, which is now Bubo scandiaca due to a genus change (Ed.: twice! as pointed out by Christopher Taylor). In general, the name printed earlier in the manuscript is given precedence in the case of subjective synonymy, but this is a rule of thumb required by the ICZN.
Another case where this principle applies is when the publication of two names, later determined to be synonyms, are on so close a date that it cannot be determined which was published earlier, the First Reviser decides precedence. With pre-20th century publications or personally published manuscripts it can sometimes be difficult to determine the exact day of publication, so synonyms published within the same month which cannot be determined to the day would fall under this ruling.
The third case where the First Reviser is necessary is when multiple spellings are used by the original author and it is unclear which spelling is considered valid. If even one of the original authors have used one of the spellings in a subsequent work, that spelling is then fixed and doesn't require a First Reviser. The rule of thumb is to fix the spelling used most often, or first in the manuscript, but this is not required. Unlike the earlier cases, the misspelling is considered to be unavailable and treated as unpublished.