There are thirteen modern constellations in the Zodiac. In modern astronomy, a constellation is a specific area of the celestial sphere as defined by the International Astronomical Union. In total, there are 88 constellations.
Astronomy and Astrology are not the same thing. Astronomy is a science while Astrology is not. As such, I'll restrict myself to the historical and modern constellations of the Zodiac.
According to Encylopedia Brittanica
Zodiac (is) a belt around the heavens extending 9° on either side of the ecliptic, the plane of the earth’s orbit and of the sun’s apparent annual path.
In historical astronomy, the zodiac is a circle of twelve 30° divisions of celestial longitude that are centered upon the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. Historically, each of these divisions were called signs and named after a constellation: Sagittarius, Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, and Scorpius.
In 1930, the International Astronomical Union defined the boundaries between the various constellations, under Eugène Delporte , who,
... drew his boundaries along vertical lines of right ascension and horizontal parallels of declination. One governing principle was that all variable stars with an established designation would remain in that constellation, as requested by the IAU’s Variable Stars committee.
"Constellations ecliptic equirectangular plot" by Cmglee, Timwi, NASA - Own work, http://svs.gsfc.nasa.gov/vis/a000000/a003500/a003572. Licensed under Public Domain via Commons.
As a result, the path of the ecliptic now officially passes through thirteen constellations: the twelve traditional 'zodiac constellations' and Ophiuchus (which was one of the 48 constellations listed by the 2nd-century astronomer Ptolemy), the bottom part of which interjects between Scorpio and Sagittarius.
Source: journeytothestars.wordpress.com
So, the 13 constellations of the Zodiac are Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius and Ophiuchus.
As seen from the (first) figure, the ecliptic also touches the edge of the constellation Cetus, though it in not usually included in the Zodiacal constellations.
Note about Precession: Because the Earth in inclined (by $23.45^{\circ}$), it rotates like a top. This is called precession, which results in a shift in the position of the constellations relative to us on Earth. The result is that the 'Signs of the Zodiac' are off by about one month.
