On July 12th, Neptune completed one full circuit around the Sun since its discovery on the night of Sept 23-24, 1846.
Neptune's discovery is one of the most remarkable stories in astronomy history. The planet's existence had been predicted due to its gravitational perturbation of Uranus's orbit, and Neptune was actually found just a degree from its predicted position. But it turns out that this remarkable accuracy was due, in some sense, to luck as much as skill. You can read the whole story in our July issue.
This is an awkward time to observe Neptune; it doesn't reach a reasonable height above the horizon until the wee hours of the morning. Moreover, it won't appear anywhere near its discovery position with respect to the stars, due to the fact that Earth is in quite a different part of its orbit. But if you want to view Neptune anyway, it's easy to do with decent telescopes and binoculars. You will need a good planetarium program or our online observing guide and charts.
Neptune is well-placed in the early evening sky starting in September, and it will appear extremely close to its discovery position among the stars from mid-October through December.
Thanks to "Bob D" for pointing out in a comment to a different online story that an argument can be made that Monday, rather than Tuesday, is Neptune's true birthday. According to Bob (I haven't checked the calculation), Monday is when Neptune is in its discovery position relative to the solar system's center of mass, which actually lies near but not at the center of the Sun.
When will Neptune really be at its discovery position in space? According to the Theory of Relativity, that's not a well-defined question. But Neptune will never again come anywhere close to its discovery position with respect to the average position of extremely distant galaxies, the closest thing we have to "absolute" position. That's because the Sun has been swept along in its rotation around the galaxy, and the galaxy has moved in its orbit around the center of the Virgo Supercluster.