Orbits have several important components, namely the period, the semi-major axis, the inclination and the eccentricity. You can only compute the eccentricity and the inclination from observations of the orbit itself over time, but the semi-major axis and the period are related mathematically. If you know one of these parameters, usually determined originally from observations, you can determine the other. It is possible to find the semi-major axis of many orbits from information tables about astronomical objects. Once you have the semi-major axis, you can find the period of an orbit.
Steps to Calculate the Period of an Orbit
Look up the semi-major axis of the orbit you want to use. Astronomical tables for planets usually list the semi-major axis as the distance from the Sun. The semi-major axes for other bodies are their distances from their centers of rotations. The semi-major axis of the Moon, for example, is its distance from Earth.
Convert the units of your semi-major axis to astronomical units. An astronomical unit is equal to the distance of the Earth from the Sun. That distance is 93,000,000 miles or 150,000,000 kilometers.
Use Kepler’s Third Law to calculate the orbit’s period from its semi-major axis. The Law states that the square of the period is equal to the cube of the semi-major axis (P^2 = a^3). In order for the units to be correct, the semi-major axis should be in astronomical units, and the period should be in years.
Convert the period into the most appropriate units. For fast-moving bodies with small orbits (like the planet Mercury or the Moon), the most appropriate unit is usually days, so divide the period in years by 365.25. Larger orbits have longer periods that you should generally measure in years.
TL;DR (Too Long; Didn't Read)
If you can not find the necessary orbital parameters in an astronomical table (this can be the case for artificial satellites and newly-discovered comets), you can try to determine semi-major axis and period with observation. You will need many observations conducted with precision over time in order to begin. There are computer and calculator programs that can determine the orbital parameters from your observations.
When checking astronomical tables for semi-major axes, try to find the value for the maximum distance between the object and the orbital center. Using the average or mean distance will only give you an approximation for the semi-major axis, based on the assumption of a circular (rather than elliptical) orbit.