The waxing crescent covers the interval between conjunction and first quarter, elongation from zero to ninety degrees, with the lit limb turned toward the Sun and therefore toward the western horizon after sunset in the northern hemisphere. Its orientation is not fixed: from the tropics the crescent appears to lie on its back, and from the southern hemisphere it points the other way, so descriptions of the crescent's direction are local rather than universal.
The phases are produced by the changing angle between the Sun, the Earth and the Moon, and the quantity that defines them is elongation, the Moon's angular distance from the Sun as seen from the Earth. The complete round from one new moon to the next is the synodic month, about 29.53 days, and it is longer than the sidereal month of about 27.32 days in which the Moon returns to the same place among the stars, because the Earth has moved along its orbit in the meantime and the Moon must travel further to catch up.
The first visibility of the crescent after conjunction is the single most consequential observation in the history of calendars, since it begins the month in the Hijri system and began it in many older ones. Whether it can be seen on a given evening depends on the elongation, the altitude above the horizon at sunset and the clarity of the air, and predicting it is a genuine problem with a substantial technical literature rather than a simple calculation.