Picture the Earth's axis extended out into space. It points at two fixed spots on the sky: the north celestial pole (almost exactly where Polaris sits) and the south celestial pole. The whole sky wheels around those two points every night as the Earth spins.
Your latitude tilts that wheel. Stand at the North Pole and the north celestial pole is straight overhead — the entire northern half of the sky circles above you and you never see anything south of the celestial equator. Stand at the South Pole and it is the mirror image. Stand somewhere in between and you see your own pole lifted up by an angle equal to your latitude, while the opposite pole sits the same angle below your horizon — hiding a cap of sky you can never reach.
Close enough to your raised pole, a star never dips below the horizon: it just circles the pole all night, every night of the year. These are your circumpolar stars. The higher your latitude, the bigger that never-setting circle.
The seasons are flipped across the equator — December is midsummer in Sydney and midwinter in London — and the night sky flips with them. The constellations you can see at all from both hemispheres (the zodiac and the equatorial band) appear at opposite times of the year. Orion rules the northern winter, from about December to March; south of the equator that same Orion rules the summer. The Milky Way's bright core in Sagittarius and Scorpius is a winter object for the north and, again, a summer showpiece for the south.
The reason is simply that it is the same Earth going around the same Sun: whichever constellations sit opposite the Sun are the ones on view at midnight, and the calendar label on that month is reversed between the two halves of the planet.
A traveller from the north who reaches Australia often does a double-take at Orion. Nothing has changed about the stars; the observer has been turned upside down relative to them. Facing the celestial equator, a northerner looks south to see Orion and a southerner looks north — so each sees the pattern rotated 180° from the other. Betelgeuse and Rigel swap corners; Orion's “sword” hangs up instead of down.
The same happens to the Moon. The face is identical, but a southern observer sees it rotated half a turn, so the “Man in the Moon” and the lit crescent tip the other way. A waxing crescent that curves like a smile from the north looks like a frown from the south. Even the Sun tracks across the northern part of the sky as seen from down south, rather than the southern part as we are used to in the north.
At 1.3° north, the Science Centre Observatory sits almost exactly on the equator, where both celestial poles lie on the horizon at once. That is the rare place where the split stops mattering: over a year, the far north and the far south of the sky both come past us. On a single clear night we can show a visitor the Big Dipper low in the north and the Southern Cross low in the south — two emblems that no single-hemisphere observatory on Earth can put in the same sky.
It comes at a price: both poles being on the horizon means the far-northern and far-southern showpieces are always low, seen through the thickest, haziest air. Wide access, never easy access. The companion page on why Singapore sees almost the whole sky works through exactly how much — and where the one small blind spot is.