The Equatorial Coordinate System
This system allows the position of a celestial object to be directly given regardless of the observer’s location. This system is analogous to the Earth's Latitude/Longitude coordinate system but is projected outwards onto the celestial sphere, it gives two-dimensional position fix as it gives no indication of the distance to an object.

The Celestial Sphere
The direction directly above the Earth's North and South poles are called the North Celestial Pole and South Celestial Pole respectively and the ring above the Earth's equator is called the 'Celestial Equator'. 

The Equatorial coordinate system projected around the Earth

Declination
The equivalent of latitude is called Declination or Dec, starting at the Celestial Equator with Declination zero degrees and increasing as you move towards the North Celestial Pole with positive Declinations, so the ring above the latitude of 51 degrees North would have Declination +51 degrees similarly the ring above the latitude of 51 degrees South would have Declination -51 degrees. 

Measurement of Declination away from the Celestial Equator

Right Ascension
The equivalent of longitude is called Right Ascension or RA. The zero point of RA is defined as the Vernal Equinox, the point on the Celestial Equator where the Sun's path crosses from South to North (the ascending node). The scale used for Right Ascension is measures in hours, minutes and seconds to give a full circle of 24 hours, this means that one hour of RA is equal to 15 degrees of longitude. From the staring point at the Vernal Equinox, RA increases with rotation to the East so that as the Earth rotates the RA passing over the observer's meridian increases. It is worth noting that R.A. hours, minutes and seconds are 15 times larger than Dec. degrees, minutes and seconds so you may see that R.A. seconds are given with one decimal place precision more than that of Dec.

Measurement of Right Ascension

Position Coordinates
The R.A. and Dec. coordinate grid is used to give a quantitative position measurement for a celestial object's position in the sky for example:

The star Betelgeuse in Orion has position coordinates of 05hours, 55minutes and 10.3053 seconds RA and +07deg, 24minutes and 25.426seconds Declination, although this is abbreviated to:

RA 05h 55m 10.3053s  Dec +07^o 24' 25.426"


The star Rigel has position coordinates of:

RA 05h 14m 32.272s  Dec -08^o 12' 05.91"

Hour Angle
Hour Angle is used to describe the angle, in hours minutes and seconds, from the observer's meridian. It's zero point is the meridian above the observer and decreases Eastwards and increases towards the West, so an object that will pass over the observer's meridian in two hours time will have an Hour Angle of -2h and an object that passed over the meridian 4hours, 10minutes and 35seconds ago will have an Hour angle of +4h 10m 35s.

Measurement of Hour Angle

Vernal Equinox
The Vernal Equinox is changing continuously due the precession of the poles, the wobble in the spin of the Earth's rotational axis, tracing out a circle in the sky every 22,000 years or so. There are standardised values given to the coordinate system in 50 year intervals to account for the change in position of the Vernal Equinox, these are called Epochs, the latest values that you may see quoted in star catalogues were given for the years 1950 and 2000 being called B1950 and J2000, the B stands for Besselian years and the J stands for Julian years, but these are another subject.

Horizontal Coordinate System
The Horizontal system is relative to an observer's location and it locates positions on the hemisphere of the sky visible by the observer with reference to the local horizon. It is another two dimensional system consisting of an Azimuth and an Elevation or Altitude but I will use the term Elevation as Altitude can be mistaken for height.

The Horizontal Coordinate System

Azimuth
Azimuth is measured in Degrees with the starting point of 0^o being at True North, the Earth's rotational pole, and increases in a clockwise direction, through East at 90 degrees, South at 180 degrees and West at 270 degrees. 

Measurement of Azimuth

Elevation
Elevation is measured in Degrees from the horizon being zero degrees and the point directly above the observer, the observer's Zenith, being 90 degrees. Objects that are currently below the horizon can have their positions described by using negative elevations starting at the horizon with zero degrees and decreasing to -90 degrees directly below the observer, the observer's Nadir.

Measurement of Elevation

Object Positions
Celestial objects change position as the Earth rotates in this system and the coordinates at a particular time will be different for observers at different locations.

Galactic Coordinate System
This system uses our Galaxy (the Milky Way) as a reference for positions of astronomical objects, other than solar system objects. It is mostly used in radio astronomy for observing Galactic objects like the supernova remnants Tau A and Cas A.

Galactic Longitude
The Sun is the centre point with Galactic Equator aligned with the Galactic plane, is a similar way to the Earth based system the Galactic system uses Galactic Latitude and Galactic Longitude with zero degrees Galactic Longitude being coincident with the Galactic centre in the constellation of Sagittarius, at R.A. 17h 45m 37.224s  Dec. -28^o 56' 10.23" (Epoch J2000), and increases Eastwards around the Galactic plane with 180 degrees Galactic Longitude being in the opposite direction to the Galactic centre.

Measurement of Galactic Longitude

Galactic Latitude
Galactic Latitude runs parallel to the Galactic plane with zero degrees being along the Galactic plane, it increases up to 90 degrees towards the North Galactic Pole, R.A. 12h 51m 26.282s  Dec. +27^o 07' 42.01" (Epoch J2000), and decreases down to -90 degrees at the South Galactic Pole, R.A. 00h 51m 26.282s  Dec. -27^o 07' 42.01" (Epoch J2000).

Measurement of Galactic Latitude