I am curious about the precision NASA and other agencies have in the space navigation. Are they always accurate to the meter in position they aim for; in mission like the recent rover landing on the Mars? How accurate are the JPL produced ephemeris/vector calculation on tracking the movement and position of Mars, planets in space and landing a rover on the surface?
The current best ephemerides available is the JPL DE440 as described in Park et al. 2021 (paywalled) and available from the JPL ftp site. This is an incremental improvement over the previous DE430 ephemeris (detailed in the freely available IPN Progress Report 42-196). The main changes are that there is a longer baseline of spacecraft position data to fit to, particularly for Jupiter where tracking of Juno has improved the accuracy considerably (by approx. 30x). Mission-specific JPL ephemerides have been created in the past such as DE424 for Mars Science Laboratory (Curiosity) and DE434 for Juno. These tend to incorporate the most recent data for the particular planetary mission destination beyond what was in the last major ephemeris release (DE421 and DE430 in this case).
For Mercury through Saturn, positions are primarily determined from tracking of the spacecraft in orbit around those planets. For the Moon, the main contribution is from Lunar Laser Ranging to the retroreflectors left on the Moon by the later Apollo missions and astronauts and the Soviet lunar rovers. For Uranus, Neptune and Pluto, the positions are almost all from Earth-based optical telescope positions, supplemented in the case of Pluto, by stellar occultation measurements. The spacecraft positions are determined through either range or Doppler measurements for the radial distance or VLBI tracking for the on-sky position (in Right Ascension/Declination). The range measurements tend to be considerably more accurate than the positional ones. From the DE440 paper the errors on range and position are approximately (I converted the angular residuals of the VLBI measurements to a distance using average distances to each planet):
|2 (MGS)/0.7 meters (MGS, ODY, MRO)
For landing rovers on Mars, knowing the position of the planet is only part of the problem as final position is going to be determined by interaction of the spacecraft with the atmosphere at the time of entry. The extent and thickness of the Martian atmosphere varies with time, both seasonally and particularly due to large dust storms. Previous landings have tend to target safe, flat areas with a large error ellipse based on reasonable assumptions of what the Martian atmosphere will be like at time of arrival.
Mars 2020 (Perseverance) differs from this by incorporating terrain feature matching software as part of the Entry, Descent and Landing enabling the descent to steer to a much more constrained and tighter landing area.