I've seen the term 1850 stats used often when describing Pokemon competitively, but I've never been sure what it means.
That's a Smogon specific term.
It refers to the Pokemon usage stats of any specific tier based on the people who are ranked 1850 and above on the Showdown ladder for any tier. The reasoning behind this is that many people who aren't "seasoned" battlers can get up the ladder by dumb luck. 1850 ranking is the cut off point since it's safe to assume that anyone who got to that point knows a significant amount of competitive battling, so their judgement on Pokemon usage is based upon competitive aspects, rather than solely personal preferences.
Example: RU stats for April 2013 (1850 stats)
This is the specific post of the 1850 stats