As the earth rotates on its axis from W to E, all the celestial bodies appear to move in the opposite direction from E to W. This means that the way the sky looks like to you now is exactly the way it looked to someone E of you some time ago, and exactly the way it will look like to someone W of you some time from now. In other words, you can't tell how far E or W you are (Longitude) without an accurate timepiece to correct for the earth's motion. On the other hand, how far north or south of the Equator a star is in the sky changes only very gradually, it is written down in the Almanac where you can look it up. The sun does move N or S throughout the year, but this is also published in the Almanac for each day, you don't need to know the precise time. So it is possible to determine your Latitude without a chronometer, even if you can't determine your Longitude. This is called Latitude Sailing.
The Arabs and probably the Chinese knew about Latitude Sailing back in medieval times, and Columbus knew about the technique (although he wasn't very good at it). I'm sure even the Ancient Greeks were familiar with it, their astronomers were, although merchant sailors weren't too eager in those days to share their trade secrets with potential competitors so history doesn't tell us for sure. Until the invention of chronometers in the middle of the 18th century, Latitude Sailing was the only form of celestial navigation available. You simply sailed N or S to the Latitude of your destination, then E or W until you hit it. Of course, if the winds didn't cooperate it could be quite inconvenient, and if you missed your destination, you could keep right on going past it!
The stars and sun all rise in the east, more or less, and roll around the sky until they cross the meridian, that imaginary line that goes from the N to S pole, directly over your head. At meridian crossing, the star is due S of you if its declination is S of your Latitude, due N if its declination is N of your Latitude. If it's declination is the same as your Latitude, then it will be directly overhead. In either case, this is as high as the star (or Sun) can get from the horizon as seen from your position. Declination is just another way of saying the Latitude of the point directly beneath the star.
Since the celestial equator is directly above the earth's equator, it can be said to have a declination of zero degrees. Now you can't see the equator to measure its altitude above the horizon, after all, it is an imaginary line; but if you could, your Lat would be 90 minus the equatorial altitude. So for example, if you saw the equator at it's highest point to be 10 deg above the horizon then your Lat would be 80 degrees. If the equator arced to 75 deg above the horizon, your Latitude would be 15 deg. Whether those would be N or S Latitudes would depend on whether that high point where equator crossed meridian to be N or S of you.
Unfortunately, the equator's altitude can't be measured with a sextant, but a star's altitude can, and we know exactly how far each star is from the equator (the declination) from the Almanac. So a simple addition or subtraction tells us where the equator is! Of course, the devil is in the details, and exactly how to do the sums depends on whether the star is N or S of you, N or S of the equator, and whether you are N or S of the equator. I'll work the easiest case, and leave the others to you as "an exercise for the student".
At LAN (Local Apparent Noon, the moment the Sun is as high above the horizon as it can get) you observe it with your sextant to be 47 deg 14.7 min above your S horizon. You look up the declination of the sun in the almanac for that day and make a rough correction for the time of day
(remember, you don't have a chronometer but you have a rough idea of your DR Longitude and the time correction for it in the almanac). Incidentally, LAN doesn't necessarily happen at twelve o'clock, but that's a topic for a different lecture. You find that according to the Almanac, the sun's declination is -18 deg 05 min. So the sun is 18 deg 05 min S of the equator, therefore, adding the two, the equator must be 65 deg 19.7 min above the S horizon. Consequently, (subtracting from 90) you must be at Lat 24 deg 40.3 min N.
Since you don't know the exact time (we're assuming you don't have a chronometer or other access to Greenwich Mean Time), you can't can't calculate exactly when LAN is going to happen. So you have to start making sextant observations sometime before , and writing down your altitudes, watching them get higher and higher until they start shrinking again (this means you've passed the exact moment of LAN). Unfortunately, the sun moves very slowly at the top of its arc and it is very difficult to exactly determine or predict the exact moment it is at its peak. Still, the LAN sight (navigators do a "noon sun" every day just to keep in shape because this is such an important backup procedure) will give you a fairly good latitude, so once you have reached the latitude of your destination, you simply sail east or west until you hit it.
When I lived in San Francisco I read about a sailor who was knocked down in a Pacific storm, severely damaging his rig, and taking out all his electronics. Without electricity, he lost GPS and time ticks, so even his celestial navigation was compromised. He was able to erect a jury mast and fly a spare jib as a lateener, and he used Latitude sailing to find the Golden Gate and safety. So this is not just an academic exercise.