Eclipses occur every year, but seeing one in one particular location on Earth is more rare. There are different types:

Solar Eclipses

Happen when the moon blocks out the sun.(moon casts a shadow onto Earth)
↳ Total: the moon perfectly blocks out the sun such that only the corona(atmosphere) of the sun is visible.
↳ Annular: the moon is further away so it does not cover the sun completely, but rather forms a ring during totality. The moon is further away because the moon's orbit is not a perfect circle.
↳ Hybrid: They are eclipses which if seen form a location on either end of the eclipse's path look annular while at a location near the center of the eclipse's path look total. Occur when the moon is not at its furthest nor its closest distance to Earth.

Lunar Eclipses

Happen when the moon moves into the Earth's shadow. When it happens it is visible everywhere on the night side of the Earth because the Earth's shadow is big enough (the moon is smaller and so its shadow is smaller and so solar eclipses are not visible everywhere on the day side).
You would think that a lunar eclipse is when the moon should be full but is not visible, but it actually becomes red. The moon does dim a lot because it is in the shadow of the Earth so the sun does not illuminate it, but a little bit of light that passed through the Earth's atmosphere does illuminate the moon. It is red because red is longer wavelength then the other visible light so it can get though obstacles(all the atoms and particles in the atmosphere) more easily than short wavelength.

A partial solar or lunar eclipse is seen when you are in a location that is close to the shadow, but not within it.

Predicting Eclipses

The Big Picture

The orbit of the Earth around the sun is defined to have 0 degrees inclination and is called the ecliptic. From our point of view here on Earth it looks like we are standing still and the sun is orbiting Earth on the ecliptic. The orbital plane of the moon has an inclination of 5 degrees relative to the ecliptic(for reference, diameter of moon and sun from Earth is ~1/2 a degree). Meaning that it intersects the ecliptic at 2 points(called nodes). These are the only times when an eclipse can(does not have to but can) happen.

For an eclipse to happen the moon has to either block out the sun(solar) or be in the shadow of the Earth(lunar). To block the sun the moon has to be between the Earth and the sun, which is what we call new moon. And to be in the Earth's shadow the moon has to be behind the Earth, which is what we call full moon.(The phases of the moon are created because the sun only illuminates the side of the moon that the moon shows the sun. So as the moon moves in its orbit we don't always see what the sun sees and so part of the side of the moon is in the dark some times creating phases).
The moon's orbital inclination like Earth's tilt does not change direction as the Earth travels around the sun. So if the points of intersection of the moon's orbit and the ecliptic happen at full and new moon then a quarter year later they will be at half moon. This is why eclipses happen somewhere on Earth every 6 months. The Earth travels around the sun slow enough that when the alignment is right for a solar eclipse then the alignment is still right for a lunar eclipse ~2 weeks later(and vice versa). An eclipse season is between 31 and 37 days long so there are always 2 (1 lunar and 1 solar or annular) and some times 3 eclipses.

So, in summary, an eclipse happens somewhere on Earth if it's a full or new moon and its orbit intersects the ecliptic.

The Moons Orbit and Eclipse Occurrence in Detail

The moon's orbital period does not fit nicely a certain number of times into Earth's orbital period around the sun. So every year the moon does 12 orbits and a partial orbit(technically the moon does 13 orbits and a little bit(here 1 orbit is when the moon is in same position relative to the background stars(this is called sidereal month)), but because the Earth orbits the sun the moon has to travel further to have the same phase again so there are only 12 and a little bit phase cycles(synodic months)). These partial orbits add up so that after 18 years and 11 and 1/3 days the Earth's year and the moon's orbital period align again. This is why every 18 years and 11 and 1/3 days the moon will have the same phase as it had on the same day 18 years and 11 and 1/3 days ago. This is called the Saros cycle.
So after 18 years and 11 and 1/3 days the same eclipses will occur again. But because of the 1/3 day the eclipses will not be visible at the same location on Earth but rather 1/3 Earth rotations(~8hours) westward(Earth rotates eastward aka from west to east).
So in theory every 3 saros cycles(~54years)(called exeligmos) the same eclipse should occur at the same location on Earth. This is true for lunar eclipses since they are visible everywhere on the night side of the Earth. But this is not the case for solar eclipses which reoccurs at a similar location but more north or south. Why?
Because the moon's orbit is not a static ellipse:

New moon to new moon=~29.5 days(synodic month)
Perigee to perigee (1 lunar orbit)=27.5 days(anomalistic month)(perigee is closest point in orbit)(anomalistic month is longer then siderial month because the perigee moves(precesses))
Node to node=~27days(draconic month)

A draconic month is not equal to an anomalistic month because the Sun's gravity(and Earth and other planet's gravity) causes the orbit of the moon to change. The moon's orbit is like a plane tilted at 5 degrees relative to the ecliptic(orbital plane of Earth). This plane slowly rotates/presses westward in the y axis(Nodal precession) and since the moon orbits eastward, the time it takes to go from a node to the same node is slightly shorter. This is also why the eclipse season(the time when everything aligns ~ every 6 months) is always a few days earlier every year. The orbital plane of the moon does one full rotation in the y axis every 18.6 years.
A node is just a point so an eclipse can occur a little bit before or after a node. If the moon goes from being more north to more south its called descending node and from south to north its called ascending node. If the eclipse occurs on a decending node, then the same solar eclipse will occur at a slightly more south location then it was 3 saros cycles ago. And for ascending node it would shift more north.

But the orbital plane of the moon does not just rotate/presses in the y axis but also in the x axis(Apsidal precession) causing the perigee(closest point in orbit) and apogee(furthest point in orbit) to rotate eastward and do a full rotation every 8.85 years. This is why both annular and total eclipses can happen(Our moon is just the right size and distance for a total eclipse to be possible this is not necessarily the case for moons of other planets).

An anomalisitc month is not equal to a synodic month not only because of Apsidal precession, but also because Earth orbits the sun so the angle the moon and Earth have to the sun changes so it takes a little bit more than one orbit of the moon around the Earth to have the same phase again.

The moon's orbit also presseses/rotates in the z-axis(Axial precession). This pressesion is the moon's axis itself making a circle. It rotates westward and completes a full turn every 18.6 years. This pressesion only affects eclipses because the moon is not perfectly round and so its shadow is'nt either.

The path of a solar eclipse is also affected by the tilt of the Earth. This means that if the path of the shadow is from south to north, then half a year later when the Earth is on the other side of the sun the path of a very similar eclipse would be from north to south. The Earth's tilt does not change over half a year, Earth is inclined in the same direction all year which causes the seasons(The Earth also precesses around its axis like the moon, but this is only noticeable over thousands of years).
That means that even though every saros cycle the path of a solar eclipse is the same just shifted north or south, and westward, over time the angle of the eclipse's pass relative to the equator will shift since a saros cycle is not just 18 years but also 11 and 1/3 days meaning that over time the eclipse will reoccur at a later and later month.

Predicting Date, Time and Place; Saros Series; Putting It All Together

Quick recap: every year there are usually at least 2 lunar eclipses and 2 solar eclipses somewhere on Earth because ~ every 6 months the alignment is right for one orbit. Every 18 years and 11 and 1/3 days the alignments are right to have the same eclipse as 18 years and 11 and 1/3 days ago just at a location 8 hours westward(saros cycle).
For lunar eclipses that means that they reoccur every 54 years and 34 days in the same locations meaning you can see one at least every 54 years, but probably more often because since half the Earth is always in the dark(which is where they are visible) and that there are 2 lunar eclipses per year the chance of seeing one every few years is very high.
For solar eclipses on the other hand, that means that every 54 years and 34 days the same eclipse will occur on the same part of the Earth, but the angle of the path relative to the ecliptic changes by ~15 degrees(because 34 days=~1 month and in 6 months a path changes by ~90 degrees so 90/6=15 degrees) (counterclockwise(CCW)), and the path will be more south if the eclipse is on a descending node or more north if on an ascending node.
A saros series is all eclipses occurring every 54 years and 34 days(exeligmos). On an ascending node the moon moves "upward" on every exeligmos making the eclipses pass move northward every exeligmos both before the moon passes the node and after it passes the node. For descending node it's the same just that the moon moves "downward" on every exeligmos and so the eclipse pass moves southward. So a saros series always starts as no eclipse then as the moon moves closer to the node it becomes partial and then total and when the moon is passed the node it becomes partial again and then becomes no eclipse and the series ends. Odd numbered saros series are ascending nodes and even numbered series are descending nodes.

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