[This question is most popular around Christmas time.]

It is first and most important to stress that the Bible is a religious
book.  The Star of Bethlehem is mentioned only briefly in the book of
Matthew.  As such Matthew's description of it may have been religious
rather than scientific.  Indeed, it has also been pointed out that the
Star story is similar to a Jewish Midrash, or moral tale illustrating
a religious point, which does not necessarily have to have any basis
in fact.  Furthermore, at the time the Bible was written the word
"star" could be used to indicate essentially anything in the sky.  The
Star of Bethlehem was almost certainly not what we understand today a
star to be (namely a ball of gas shining by interior thermonuclear
fusion).

Nearly any spectacular sky phenomenon (comet, supernova, nova, etc.)
has been identified as the Star of Bethlehem at one time or another,
but recent interest has focussed on conjunctions of various planets,
possibly in auspicious constellations.  Two examples are the
following:

Michael Molnar has found that there was an double occultation of
Jupiter in March and April of 6 BC in Aries that would have been
calculable even by the means available to astrologers (which the Magi
were) and that would have been of high significance in magian
astrology (which differed somewhat from astrology of the modern era).
However it would have been invisible, taking place in daylight.  Thus
there is a perfectly good explanation as to why Herod's courtiers had
not seen it, but "wise men from the East" knew all about it.  The
occultation also provided a neat explanation of why the star was seen
over Bethlehem---from Jerusalem, the second occultation's azimuth was
close to the direction of the town.  Molnar also points out that the
Romans regarded the horoscope of Jesus as a royal one.

And for a small commentary on one of Molnar's points, see my paper
with Steve Fossey in The Observatory in 1998 or at
<URL:http://www.star.ucl.ac.uk/~mmd/star.html>.

On 3 May 19 BC, the planets Saturn and Mercury were in close
conjunction, within 40 minutes of arc of each other. Then Saturn moved
eastward to meet with Venus on 3 June 12 BC.  During this conjunction
the two were only 7.2 minutes of arc apart.  Following this
conjunction, on 3 August 12 BC, Jupiter and Venus came into close
conjunction just before sunrise, coming within 4.2 minutes of arc from
each other as viewed from earth, and appearing as a very bright
morning star. This conjunction took place in the constellation Cancer,
the "end" sign of the Zodiac. Ten months later, on 2 June 17 BC, Venus
and Jupiter joined again, this time in the constellation Leo. The two
planets were at best 6 seconds of arc apart; some calculations
indicate that they actually overlapped each other. This conjunction
occurred during the evening and would have appeared as one very bright
star. Even if they were 6 seconds of arc apart, it would have required
the sharpest of eyes to split the two, because of their brightness.

(Some of this information is adapted from a longer article at
<URL:http://sciastro.net/portia/articles/thestar.htm>.  There is also
other pertinent information at this site regarding the astronomy
during that time.)