A planet can be said to "transit" a sign in the sense of passing through that sign (though due to retrogression a planet may enter a sign, leave it then re-enter it). Since, as Moog pointed out, the tropical zodiac is (currently) offset from the sidereal zodiac by about 24°, the date that a planet enters a sign is earlier in the tropical zodiac than it is in the sidereal zodiac. For example, Mars enters Capricorn on November 18, 2012, according to the tropical zodiac, but it enters Capricorn on December 18, 2012, according to the sidereal zodiac. Due to retrograde motion, the time that a planet spends in a sign may be different in the two zodiacs.
The more usual use of "transit" is to denote the period during which a planet forms an aspect (square, etc.) with some other planet (this is called a "world transit") or with some natal planet (this is called a "personal transit").
A particular aspect between two moving planets begins when the angle between those planets differs from the exact value for that aspect by less than the orb chosen for that aspect (e.g., 7° for square aspects). For example, Venus is currently square to Mars. This aspect began on September 11, 2012, and will end on October 12, 2012, thus lasting 32 days. The angle between the two planets is independent of the zodiac used (because the angle is the
difference of the ecliptic longitudes of the planets), so the time that such an aspect begins is also independent of the zodiac used. Similarly the time that the aspect ends is zodiac-independent, and thus the duration of the transit is also. Thus world transits do not depend on choice of zodiac.
The case is different for personal transits, since the angle between a transiting planet and a natal planet is not calculated for a specific time, but must take into account the difference between the birth date and the current date (the date at which the transit is occurring). For this reason, the start and end of personal transits (and thus their duration)
does depend on the zodiac used, and the extent of the difference according to the two zodiacs further depends on the person's age and on the two planets forming the aspect. This is explained in more detail in an article
The Influence of the Tropical and Sidereal Zodiacs on Personal Transits.