What is the standard coordination system used to describe the location at a celestial object approximately due east and at an angle of 10 degrees above the horizon?
The coordinate system for an object in the sky measured relative to the horizon is known as altitude-azimuth. The altitude is the angle above the horizon (10 degrees in your case), and the azimuth is the angle around the horizon, usually measured from North. (Due east would be 90 degrees azimuth in your example.)
There isn't one, because there cannot be one: which celestial object you see towards the east at 10 degrees above the horizon depends on:
Your position on Earth
The time of day
The time of year
Because of Earth's rotation and motion around the sun, the positions of celestial objects change.
Among themselves, stars keep (for our purpose here) their relative position, and astronomical coordinate systems make use of that fact. The celestial sphere on which we project the stars rotates roughly around the North Star.
Angular distance from the North Star (or more correctly, from the celestial equator) is called declination.
As you know from Earth's coordinate system, the position of the Null Meridian is arbitrary. On Earth, it's in London, in the sky, it's in the constellation pisces. This is called right ascension.
More information: wikipedia.
Note also that, depending on the application, other coordinate systems may be used in astronomy (such as galactic or heliocentric).