The mean position of a celestial object is the average position it takes over the course of a year, based on mathematical models. This may differ from it's true position judged from direct observation.
A good example of the concept of 'mean' vs 'real' is the Sun's daily passing over the local meridian ('high noon', easily measured with a sundial). One might think the sun passes over this meridian once every 24 hours. In fact, the time can vary from 24 hours by as much as 30 minutes. Only when taken in average over a year is the time, the 'mean time', 24 hours. 
The concept also applies to motion of bodies along the ecliptic path through the Nakṣatras over the course of a year. The 'mean sun' for example, is a model of the sun in which it moves along the ecliptic at a constant speed, of approximately 1° per day (365.25 days in a year, 360 degrees in a circle). The 'real sun' moves along this path at a variable speed over the year. And so, the 'mean position' of the sun differs from the 'true position' of the sun. 
Further complicating matters is the geocentric model of epicycles, in which planets had two orbits: one around a 'mean planet', the mean planet itself orbiting the Earth.
- The Amateur Astronomer's Introduction to the Celestial Sphere, William Millar, Cambridge University Press, 2006. pgs 82-84, 270