Today, 2017 September 3, is Julian day 2,458,000. The last three digits just rolled over to all zeros. Granted, it's not as dramatic as rolling over to Julian day 3,000,000 (which won't happen for nearly 15 centuries), but it's something.
In business, the term "Julian date" is often misapplied to the day of the year, as in, today is the 246th day of 2017. But the actual Julian day number is a count of days since the start of the year 4713 BCE as reckoned by the Julian calendar. More precisely, it's number of days since noon Greenwich Mean Time on the start of January in the year 4713. Keeping track of events in terms of Julian day numbers make it easy to work out the number of days between the events by simple subtraction.
When astronomers calculate the positions of planets, moons, asteroids, and spacecraft, it’s usually much easier to work in time units of fixed length, which rules out months and years, and given the time spans in question on the scale of the solar system they prefer to work in days rather than shorter units. For these purposes a day is always 86,400 seconds long. Leap seconds are occasionally added to civil timekeeping, mainly because the Earth's rotation is very gradually slowing down, but since they would hugely complicate the computations we're talking about, they're ignored.
Under the Julian day system, a specific time is given by a decimal fraction appended to the day number. For example, 2,456,293.520833 was the Julian date or Julian day number for thirty minutes past midnight GMT on January 1, 2013 (with the date reckoned by the current Gregorian calendar).
The idea of using 4713 BCE as the starting point was proposed by French historian Joseph Scaliger in 1583, which was when the Gregorian calendar was just being introduced and had people thinking about such things. Scaliger wanted to get rid of the complication of dating much of ancient history backwards from the supposed birth of Jesus. He might have simply shifted the starting point by a nice round 10,000 years (so 2017 would be the year 12,017, for example), but given the complications of the switch from the Julian to the Gregorian calendar with an accompanying shift of New Year's Day (formerly March 25, believe it or not), this wouldn't be quite as simple as it sounds.
What makes 4713 BCE interesting is that it’s the last year when three historically interesting cycles were all starting over at the same time under the Julian calendar. These were (1) the 15-year indiction cycle for revaluation of land for property-tax purposes in the late Roman Empire, (2) the 19-year Metonic cycle after which full moons return to the same sequence of calendar dates, and (3) the 28-year solar cycle, after which the Julian calendar repeats with respect to days of the week. (If you’re missing a calendar for a given year, you can just recycle one for 28 years earlier.)
Scaliger's proposal never caught on with historians, possibly because it would in practice be confusing. (It would be much easier to shift the starting point back by 10,000 years for both Julian and Gregorian dates. It seems an obvious idea, but I've never seen anyone else suggest it. If anyone decides to adopt it, feel free to calling Gradian dating.) But in 1849 astronomer William Herschel proposed adopting a variant idea, namely assigning a number to each day counting from the starting date Scaliger suggested. A few decades later astronomers adopted Herschel's idea except for using Greenwich as the reference point. Herschel had wanted to reckon days from noon mean solar time in Alexandria, the home town of Ptolemy, a brilliant mathematical astronomer of the classical era.
Why start the day at noon? Doing so is a longstanding tradition in astronomy. As long as the sky is clear and you're not too near a pole at the wrong time of year, it's easy to establish local noon by checking for maximum height of the sun over the horizon, and starting the day at noon avoids having the date change interrupt your nighttime viewing -- at least provided you're on the side of the Earth centered on the longitude of Greenwich -- which is after all when astronomers in Europe do their observations of the stars. For people in the U.S. eastern time zone, the new Julian day starts at 7 in the morning Eastern Standard Time, or 8 a.m. Eastern Daylight Time.
(Revised a bit 2017 December 1 in an attempt to add clarity to the last paragraph.)