A basic requirement for studying the heavens is determining where in the sky things are. To specify sky positions, astronomers have developed several coordinate systems. Each uses a coordinate grid projected on the Celestial Sphere, in analogy to the Geographic coordinate system used on the surface of the Earth. The coordinate systems differ only in their choice of the fundamental plane, which divides the sky into two equal hemispheres along a great circle. (the fundamental plane of the geographic system is the Earth's equator). Each coordinate system is named for its choice of fundamental plane.
The Equatorial coordinate system is probably the most widely used celestial coordinate system. It is also the most closely related to the Geographic coordinate system, because they use the same fundamental plane, and the same poles. The projection of the Earth's equator onto the celestial sphere is called the Celestial Equator. Similarly, projecting the geographic Poles onto the celestial sphere defines the North and South Celestial Poles.
However, there is an important difference between the equatorial and geographic coordinate systems: the geographic system is fixed to the Earth; it rotates as the Earth does. The Equatorial system is fixed to the stars, so it appears to rotate across the sky with the stars, but of course it is really the Earth rotating under the fixed sky.
The latitudinal (latitude-like) angle of the Equatorial system is called Declination (Dec for short). It measures the angle of an object above or below the Celestial Equator. The longitudinal angle is called the Right Ascension (RA for short). It measures the angle of an object East of the Vernal Equinox. Unlike longitude, Right Ascension is usually measured in hours instead of degrees, because the apparent rotation of the Equatorial coordinate system is closely related to Sidereal Time and Hour Angle. Since a full rotation of the sky takes 24 hours to complete, there are (360 degrees / 24 hours) = 15 degrees in one Hour of Right Ascension.
The equatorial coordinates for deep-sky objects and stars do not vary appreciably over short durations of time, since they are not affected by the diurnal motion (the daily apparent rotation of the sky around the earth. However, note that this takes 1 sidereal day, as against 1 solar day). They are suitable coordinates for making catalogs of stars and deep-sky objects (note that Galactic Coordinates also work well, but are cumbersome to use from an earth point-of-view). However, there are effects that cause the RA and Dec of objects to vary over time, namely Precession and nutation, and proper motion, the latter being even less important. Equatorial coordinates are thus generally specified with an appropriate epoch, to account for precession. Popular epochs include J2000.0 (Julian Year 2000) and B1950.0 (Besselian Year 1950).
The Horizontal coordinate system uses the observer's local horizon as the Fundamental Plane. This conveniently divides the sky into the upper hemisphere that you can see, and the lower hemisphere that you can't (because the Earth is in the way). The pole of the upper hemisphere is called the Zenith. The pole of the lower hemisphere is called the nadir. The angle of an object above or below the horizon is called the Altitude (Alt for short). The angle of an object around the horizon (measured from the North point, toward the East) is called the Azimuth. The Horizontal Coordinate System is sometimes also called the Alt/Az Coordinate System.
The Horizontal Coordinate System is fixed to the Earth, not the Stars. Therefore, the Altitude and Azimuth of an object changes with time, as the object appears to drift across the sky. In addition, because the Horizontal system is defined by your local horizon, the same object viewed from different locations on Earth at the same time will have different values of Altitude and Azimuth.
Horizontal coordinates are very useful for determining the Rise and Set times of an object in the sky. When an object has Altitude=0 degrees, it is either Rising (if its Azimuth is < 180 degrees) or Setting (if its Azimuth is > 180 degrees).
The Ecliptic coordinate system uses the Ecliptic for its Fundamental Plane. The Ecliptic is the path that the Sun appears to follow across the sky over the course of a year. It is also the projection of the Earth's orbital plane onto the Celestial Sphere. The latitudinal angle is called the Ecliptic Latitude, and the longitudinal angle is called the Ecliptic Longitude. Like Right Ascension in the Equatorial system, the zeropoint of the Ecliptic Longitude is the Vernal Equinox.
What do you think such a coordinate system would be useful for? If you guessed charting solar system objects, you are right! Each of the planets (except Pluto) orbits the Sun in roughly the same plane, so they always appear to be somewhere near the Ecliptic (i.e., they always have small ecliptic latitudes).
The Galactic coordinate system uses the Milky Way as its Fundamental Plane. The latitudinal angle is called the Galactic Latitude, and the longitudinal angle is called the Galactic Longitude. This coordinate system is useful for studying the Galaxy itself. For example, you might want to know how the density of stars changes as a function of Galactic Latitude, to how much the disk of the Milky Way is flattened.