NGC 3680 is an open cluster located about 3600 light years (1.1 kpc) away [1] in the constellation Centaurus. Open clusters are groups of stars that formed within the plane of our galaxy from the same nebula at the same time and that have remained gravitationally bound since their formation. They basically look like extra-dense concentrations of stars, all of which have the same brightness. I did not really find a satisfactory explanation for why they are called "open", but I think it's because they don't seem to have any distinct boundaries. The other types of clusters are globular clusters, which look like spheres or globes and which therefore have names that make more sense than "open clusters".
Anyway, NGC 3680 itself is not a terribly impressive-looking open cluster. It basically looks like a slightly more dense concentration of stars spread over a region about two-thirds the size of the Moon [1]. Only 100 stars have been identified as associated with the cluster, which isn't really all that much compared to other clusters [1]. It's also not particularly close or particularly bright. Unlike some of the open clusters in the Messier Catalogue, it can't be seen without a telescope [2]. So, in many respects, it doesn't really stand out. However, the one very interesting thing about NGC 3680 is its age.
Before I discuss the age of NGC 3680 specifically, I should describe how astronomers determine the ages of stars more generally. The age of the Sun itself has been measured by looking at the radioactive decay of elements in meteorites that would have formed at the same time or immediately after the Sun did. However, we can't apply this method to other stars because we don't have meteorites from those star systems. If a star is very young, we would see it within the nebula that it formed out of or see it surrounded by a protoplanetary disk of gas or dust. Otherwise, it's hard to say exactly how old an individual star is.
However, astronomers can more easily measure the ages of stars within a group of stars that all formed at the same time, like the stars in an open or globular cluster. What astronomers do is look for all of the stars that fuse hydrogen into helium in their cores like the Sun. In a plot of brightness versus temperature, these stars will lie in a straight line running from the low brightness / low temperature corner to the high brightness / high temperature corner of the plot. This line is called the main sequence. The hottest and brightest hydrogen-fusing stars, which are also the bluest, will die first, followed by not quite as bright and not quite as hot stars, which are sort of whitish in color, followed by yellowish stars in the mid-range like the Sun, and then followed by the fainter, cooler, red stars. This means that, when astronomers create plots of brightness versus temperature for all of the stars within a cluster, they will see the main sequence (or, in other words, that line describing the relation between brightness and temperature for Sun-like stars) shrinking in length as the stars get older. Astronomers just have to identify which stars are missing from the main sequence to figure out how old a cluster of stars is.
So, back to the age of NGC 3680. It's estimated to be about 1.4 billion years old [3]. This is a relatively high age for an open cluster, although some people still referred to as an "intermediate age" cluster because it is possible to find a few more open clusters that are older. An age of 1.4 billion years may not sound that old at first given that the Sun is about 4.5 billion years old. However, the age of the stars in NGC 3680 is not what impressed me. Instead, I was impressed by the fact that the cluster as a whole stayed together for 1.4 billion years.
Open clusters orbit the center of the Milky Way within the galaxy's plane. Because they aren't very tightly bound gravitationally and because they are passing by and interacting with other stars in the plane of our galaxy, they tend to get destroyed after a few hundred million years [4]. A lot of open clusters get stretched apart by tidal forces within the Milky Way until they no longer resemble clusters. However, NGC 3680 has somehow managed to avoid being destroyed this way.
To put this into a different context, the Sun completes an orbit around the center of the Milky Way about once every 225 to 250 million years. Both NGC 3680 and many other clusters are located at about the same distance from the center of our galaxy as the Sun, which means that they take about the same amount of time to orbit the center of our galaxy. Other clusters will survive for about a couple of orbits before being gravitationally shredded, but NGC 3680 has somehow managed to survive for about six orbits.
Anyway, astronomers have spent a lot of time looking at NGC 3680 specifically because they know it's one of the few places where they can easily find a lot of stars that are 1.4 billion years old. This is important for astronomers who would like to build detailed models of how stars change into red giants and then die as either planetary nebulae or supernovae and how they create and expel elements heavier than hydrogen and helium in the process of dying. It's relatively easy to find a bunch of clusters less than a few hundred million years old where astronomers can see what happens when the largest and brightest stars evovle into red supergiants and die, but it's a bit harder to find older clusters to understand what happens when the not-so-bright and not-so-large stars change into red giants and then die. Since astronomers know that they can find 1.4 billion year old stars in NGC 3680, including a lot of not-so-bright and not-so-large dying stars, they have spent a lot of time looking at NGC 3680 to understand these stars specifically [3,5,6,7], and that is why NGC 3680 is particularly important in professional astronomy.