When watching the sky, on any given clear night, we see the stars appear to revolve around a fixed point in the Northern hemisphere. Conveniently for us, that point is nearby the star Polaris, the north star, but the point itself is directly overhead of our geographic north pole. Astronomers call that point "the North Celestial Pole" (NCP). There's a similar spot that the stars appear to revolve around in the south, visible only if you're south of Earth's equator. We call that point "the South Celestial Pole" (SCP) and it lies directly above the geographic south pole of the Earth. Just as the Earth's equator splits the planet into northern and southern hemispheres, so to does "the Celestial Equator" (CE) split the sky. It lies directly over the Earth's equator. Where these points are depends on the orientation of the Earth's rotational axis relative to the stars.
The earliest records of people noting the zodiac constellations go back to the Sumerians ~4500 years ago. In their day, very few constellations were named, but among the noteworthy ones were ones that the Sun, Moon, and other planets (yes, in those days the Sun & Moon were considered planets as they "wandered" relative to the "fixed stars") appeared to travel through. In Sumer the zodiacal constellations were Taurus, Leo, Scorpius, and Aquarius, which, on a clear enough night probably look a bit more like what they're supposed to than a lot of other constellations (except Aquarius IMO).
But it's not their fidelity to form which called their attention to the Sumerians. It was the fact that in their day those were the constellations that the Sun stood against during the equinoxes and the solstices.
We call the particular path that the Sun appears to take relative to the stars "the ecliptic". It gets that name because when the full Moon, or new Moon, crosses that path, we get a lunar or solar eclipse. The Sun appears to move along this path because the Earth is orbiting around it, so it lines up with different background stars. In this way the ecliptic also traces out the plane that the Earth's orbit lies in. Because the Earth's equator is offset by 23.5 degrees from Earth's orbital plane, the ecliptic is offset by 23.5 degrees from the celestial equator.
This makes it so that there are two points where the Sun crosses the celestial equator, the spring (vernal) and fall (autumnal) equinoxes. On those days, the Sun is on the circle that splits the sky in half, rising due East, and setting due West. The center of the Sun is above the horizon for half the day and below for half the day - no matter where you are on the Earth. So we call that position the equinox for "equal night(time and day time)". Well, near the poles it appears to just circle around the horizon for 24 hours, and refraction of the Sun's light through the atmosphere makes it appear above the horizon for a little longer than a straight line path would make it, so a twelve hour day is usually before the proper spring equinox and after the proper fall equinox by a little bit.
After the spring equinox, the Sun gradually moves further north along the ecliptic, eventually getting 23.5 degrees north of the celestial equator, its northern most position in the sky, where, in the Northern hemisphere it is highest in the sky at mid-day, and stays above the horizon for the longest time of the year. This is our summer solstice.
After the fall equinox, the Sun gradually moves further south along the ecliptic, eventually getting 23.5 degrees south of the celestial equator, its southern most position in the sky, where, in the Northern hemisphere it is lowest in the sky at mid-day, and stays above the horizon for the shortest time of the year. This is our winter solstice.
As the Sun approaches the solstices, the rate at which it moves north or south decreases to a point where that motion would be undetectable by people living in classical or ancient times. Relative to the stars, the Sun appeared to stand still - sol stasis.
If the Earth's axis of rotation kept pointing in the same direction, then which star was the north star, and which constellation was behind the Sun during the solstices and equinoxes would be the same indefinitely. The Earth, however precesses (wobbles) like a top due to the tugs of the Sun and Moon on it through gravity. With a spinning top, you'll see that the axis it spins about will itself revolve about a small circle. For a toy top, this precession takes a second or so to complete - maybe less. For the Earth, it takes 26000 years.
Now the Babylonians, the Sumerians' cultural successors, were the ones to add in the other zodiacal constellations that we know today, and they did it around 3000 years ago. They divided the zodiac up into twelve sections, because over the course of a year, the Moon's phases cycle through from new moon to new moon a little over 12 times - so the Sun would migrate through nearly one of those constellations over the course of a month. This division is a bit arbitrary. The Egyptians, for example were dividing the zodiac up into 36 sections which we call decans (10 degrees wide each) during the same time period, and other cultures chopped it up differently as well.
By the time the Babylonians were involved, the Sun no longer showed up in Taurus on the spring equinox, but instead was in Aries. The Babylonians loved to use astronomy to make predictions - not only of the motions of the stars and planets, but also of more terrestrial affairs. So it's during this time that "astrology" comes to fore and when the dates that go with Sun-sign astrology were established. They're roughly the days, if you lived 3000 years ago, when the Sun would be in those signs, if the constellations were evenly split across the zodiac.
Those turn out to be rather big ifs.
By 2000 years ago, the spring equinox position of the Sun had moved into Pisces, where it is today, although it is approaching (in a few hundred years) Aquarius (hence the song from Hair and all of your New Age "Aquarian" named things.) So now, you're about a sign and a half off from what the newspaper astrologers tell you your birth sign is. About 100 years ago, the astronomical community divided up the sky into standardized constellations, so that everywhere in the sky was part of one constellation. This, along with precession, makes it so that the Sun travels through 13 constellations (the other one is Ophiuchus, the Serpent Holder) over the year and doesn't do so evenly. The Sun spends a rather long time against the background of Picses and Virgo, and not very long at all against Ophiuchus and Scorpius.
Astronomers still call the position of the spring equinox "The first point of Aries", and for the Romans and a number of other cultures, the calendar year used to begin there. Later we began starting the year after the winter solstice, so anyone hanging onto the Spring equinox as the start date was labeled an "April Fool". On the summer solstice, the Sun lies on the Taurus side of the Taurus Gemini border, but in Babylonian times it would have been in Cancer, and directly above the latitude line that we still call the Tropic of Cancer. On the winter solstice, the Sun is in Sagittarius, but in Babylonian times it would have been right in the middle of Capricornus and above the latitude line that we call the Tropic of Capricorn.