These are the times when eclipses happen: a solar eclipse coinciding with the new moon as it obscures the Sun, and a lunar eclipse when the Earth's shadow falls on the full moon.
But we do not experience eclipses every two weeks. This is because the plane in which the Moon orbits the Earth is inclined by about 5 degrees to the plane of the ecliptic (in which the Earth orbits the Sun). The two points where the Moon's orbit crosses the plane of the ecliptic are called the nodes. The northward-crossing one is known as the ascending node, whilst the one on the opposite side of the Earth, where the Moon crosses toward the south, is the descending node.
Roughly twice a year the line of nodes (the straight line passing through both nodes) coincides with the Sun-Earth line, meaning the three bodies can fully align and eclipses are possible. This period is known as an eclipse season. It lasts between 31 and 38 days, depending on the apparent sizes and speeds of the Sun and Moon, which vary because of the eliptical orbits of the Moon and Earth. Since the lunar month is slightly shorter, at 29.5 days on average, it is usual to see two eclipses - one solar and one lunar - in an eclipse season, but roughly every seven years a third eclipse is possible.
There is another important factor to consider: the Moon's inclined orbit shifts, rather like the rim of a spinning coin just before it comes to rest. This means that the line of nodes slowly rotates in the opposite direction to the Moon's movement, an effect called the regression of the nodes. The result is that the period between alignments of the Sun with the node line is 173.3 days, and with a particular node is 346.6 days, called an eclipse year. Since this is a little shorter than a calendar year, it follows that most years there will be two eclipse seasons, but every nine years or so there will be a third.
So, if you were to win an astronomers "trolley dash" which allowed you to travel to see as many solar and lunar eclipses as possible in one year, what might your score be? Generally there will be two seasons, and although sometimes there may only be one eclipse in a season, it is more likely that each will contain two, ie four in total. Roughly every seventh year there will be a three-eclipse season, giving five in all, and every ninth year a third season may produce six eclipses. When these two cycles coincide exactly there will be two two-eclipse seasons and a three-eclipse season - a grand total of seven eclipses. This last took place in 1973, and will not happen again until 2206!