E. G. Richards says in Mapping Time, page 221, that references to any sort of calendar in the books of the Old Testament originating from before the Babylonian Exile are rare. Etymologies strongly suggest that it was tied to the seasonal year with a year that began in the autumn and was thus probably lunisolar, with months intercalated from time to time. Richards says it is likely that the start of the months was signalled by the first sighting of the new moon. There is some evidence that whenever it was seen that the barley would not be ripe by the 16th of Abib (or Nisan), an extra month was intercalated to ensure that a sheaf of barley could be sacrificed to God on the day after the Passover. The names of only four months are mentioned: Ethanum, Bul, Abib, and Ziv.
During the Babylonian Captivity the Jews adopted the Babylonian calendar, which was also lunisolar. Except during the Seleucid period (when the Seleucid calendar was used), they continued to use this until the Diaspora in 70 CE. Each year had twelve lunar months and a thirteenth was intercalated as necessary. The year started in the autumn and the day at sunset.
Other calendars were used by some Jewish sects from time to time. The apocryphal Books of Enoch and of Jubilees, dating from Maccabean times in the second century before Christ, mention a solar year of 364 days containing exactly 52 weeks. This year is divided into four quarters each with three months of 30, 30 and 31 days. It appears that the Essenes used a strictly lunar calendar.
The modern Jewish calendar uses the Metonic cycle for reconciling the solar and lunar years. This site provides the names of the Jewish months now in use, little different from those adopted during the Exile.
Sometimes it seems impossible to coordinate biblical regnal periods for Israelite kings and Judahite kings. Wikipedia explains that the Israelites used the 'non-accession year' method, which states that any part of a year that a king reigns is counted as a full year to that king. The Judahites used the 'accession year' method, in which the first part of the year, which is the accession year, is not counted, but the last one is, regardless of their lengths.