Hi Joe,
Scientists can predict solar eclipses far in advance by using our knowledge about the orbit of the Earth around the Sun and the orbit of the Moon around the Earth.
Lunar eclipses can only happen at full Moon, when the Earth is between the Sun and the Moon. Solar eclipses can only happen at new Moon, when the Moon is located between the Sun and the Earth. However, it is a bit more complicated that this! We don't actually have solar eclipses once a month, at new Moon. We also don't have a lunar eclipse once every month at full Moon, either.
The reason why eclipses don't occur every month is that the plane of the Moon's orbit around the Earth is inclined at a slight angle (about 5 degrees) to the plane of the Earth's orbit around the Sun. The plane containing the Earth's orbit around the Sun and the plane containing the Moon's orbit around the Earth intersect along a line called the line of nodes. An eclipse can only occur if the line of nodes points towards the Sun, and if the Moon lies on or very close to the line of nodes. Normally full Moon and new Moon occur when the Moon is above or below the plane of the Earth's orbit around the Sun. However, if the Moon is on the line of nodes and the line of nodes points towards the Sun, the Earth, Sun and Moon are properly aligned for an eclipse.
Here is a pdf file from the class I am teaching about the solar system showing a picture of the line of nodes and the proper conditions for eclipses. http://delta.physics.uiowa.edu/~kms/lectures_pdf/sw_eclipse_notes.pdf