What two coordinates are used to locate celestial objects?

In astronomy more than just one coordinate system is used. Each of these have a distinct purpose. For general use, most astronomers use Right Ascension and Declination. These are roughly equivalent to Longitude and Latitude respectively. The celestial equator is the same as the Earth' equator but then projected in the sky. The zero meridian is equivalent to the position of the sun on 21th March 12:00 GMT (per definition).

The definition used above is useful but not really intuitive when you start with astronomy. People tend to look more to the angle of an object above the horizon. That can be used. Then the second coordinate is the azimuth angle, or the angle from north. However, because of the diurnal motion of the Earth this position is not constant.

Another coordinate system used is the galactic system. Then the position of the centre of our galaxy (in the constellation of Ophicius) is used as the zero position. But this is mainly used by people interested in galaxy astronomy.

One can also use our solar system as a reference frame. Then the equator of the sun is the equator and all the planets move roughly on this line, aka the equinox. This is mainly of interest when you want to calculate planetary movements in the sky.

Conclusion: there is not a single coordinate system, but the first position system is the most widely used. Directly followed by the second.