You can actually find one particular Earth satellite on most clear nights and sometimes during the day with the naked eye... the Moon.
Most LEO, Low Earth Orbit satellites generally travel with the rotation of the Earth, from west to east, at an inclination that closely resembles their latitude of departure from the Earth unless a specific orbital inclination is required. This is due to the expenditure of fuel that is required to alter the orbital plane or defeat the rotational energy of the Earth. As a matter of fact, some launch sites are designed to be closer to the equator to take advantage of the rotational energy of the Earth thus providing added energy for the satellite. I.e., the closer a launch site is to the equator, such as Sea Launch or French Guiana, the more you take advantage of the Earth's rotational energy for low inclination orbits. Its less expensive.
A satellite in geosynchronous orbit would, (if you could see it), appear to wander north and south over a specific longitude on the Earth at constant altitude as it races with the rotation of the planet. These satellites are typically inserted into equatorial or low inclination orbits that are very far out in space and not visible with the naked eye. Kepler's Laws of Planetary Motion requires a 35,790 km high orbit based on a 23h56m4s day.
A geostationary orbit is a circularized geosynchronous orbit and may require what is called "station keeping" if the orbital inclination is not zero, (over the equator), and maneuvering is required. This satellite, usually a communications or meterological type, appears to hover over a specific spot on the earth.
There are also sun synchronous orbits, walking orbits, Hohmann transfer orbits, and polar orbits the latter of which are unable to elicit any help from the Earth's rotation due to their high orbital inclinations, and must provide all the energy to achieve their orbital objectives.
Since light is required to see most LEO satellites... but not too much light, they can usually be seen just after sunset and just before sunrise.