What are the key coordinate systems used in Astronomy, and where are they used?
Navigating the solar system, you'd probably track your probe using Earth-based radio telescopes in the equatorial ICRF, though you might describe the overall trajectory in terms of the ecliptic.

Navigating at sea, you might measure the Sun's position in horizontal coordinates at a time in GMT to determine your own geographic coordinates.

Practical uses in backyard astronomy:


horizontal: telling someone where to look relative to an object on the horizon; pointing a telescope on an altitude/azimuth mount
equatorial: finding or plotting an object in a star atlas; choosing atlas pages to use on a given night; pointing a telescope on an equatorial mount
ecliptic: importing comet or asteroid orbital elements into planetarium software; planets are easiest to observe when their longitude differs most from the Sun's
galactic: low latitudes have more Milky Way star clusters and nebulae; high latitudes have more external galaxies
supergalactic: pondering the distribution of galaxy clusters?