A full calendar system has a different calendar date for every day. Thus the week cycle is by itself not a full calendar system; neither is a system to name the days within a year without a system for identifying the years.
The simplest calendar system just counts time periods from a reference date. This applies for the Julian day or Unix Time. Virtually the only possible variation is using a different reference date, in particular one less distant in the past to make the numbers smaller. Computations in these systems are just a matter of addition and subtraction.
Other calendars have one (or multiple) larger units of time.
Calendars that contain one level of cycles:
week and weekday – this system (without year, the week number keeps on increasing) is not very common
year and ordinal date within the year, e.g., the ISO 8601 ordinal date system
Calendars with two levels of cycles:
year, month, and day – most systems, including the Gregorian calendar (and its very similar predecessor, the Julian calendar), the Islamic calendar, the Solar Hijri calendar and the Hebrew calendar
year, week, and weekday – e.g., the ISO week date
Cycles can be synchronized with periodic phenomena:
Lunar calendars are synchronized to the motion of the Moon (lunar phases); an example is the Islamic calendar.
Solar calendars are based on perceived seasonal changes synchronized to the apparent motion of the Sun; an example is the Persian calendar.
Lunisolar calendars are based on a combination of both solar and lunar reckonings; examples include the traditional calendar of China, the Hindu calendar in India, and the Hebrew calendar.
The week cycle is an example of one that is not synchronized to any external phenomenon (although it may have been derived from lunar phases, beginning anew every month).
Very commonly a calendar includes more than one type of cycle, or has both cyclic and acyclic elements.
Most calendars incorporate more complex cycles. For example, the vast majority of them track years, months, weeks and days. The seven-day week is practically universal, though its use varies. It has run uninterrupted for millennia.
Solar calendars assign a date to each solar day. A day may consist of the period between sunrise and sunset, with a following period of night, or it may be a period between successive events such as two sunsets. The length of the interval between two such successive events may be allowed to vary slightly during the year, or it may be averaged into a mean solar day. Other types of calendar may also use a solar day.
Not all calendars use the solar year as a unit. A lunar calendar is one in which days are numbered within each lunar phase cycle. Because the length of the lunar month is not an even fraction of the length of the tropical year, a purely lunar calendar quickly drifts against the seasons, which don’t vary much near the equator. It does, however, stay constant with respect to other phenomena, notably tides. An example is the Islamic calendar. Alexander Marshack, in a controversial reading, believed that marks on a bone baton (c. 25,000 BC) represented a lunar calendar. Other marked bones may also represent lunar calendars. Similarly, Michael Rappenglueck believes that marks on a 15,000-year-old cave painting represent a lunar calendar.
A lunisolar calendar is a lunar calendar that compensates by adding an extra month as needed to realign the months with the seasons. An example is the Hebrew calendar which uses a 19-year cycle.