I want to offer the user the possibility to introduce dates that are incomplete (for example just the year 2005, or Jan 1998) but also complete dates with year, month, day.
Storing parts of dates depends on the sql mode MySQL is running in. Please read http://dev.mysql.com/doc/refman/5.1/en/server-sql-mode.html, especially about invalid dates in http://dev.mysql.com/doc/refman/5.1/en/server-sql-mode.html#sqlmode_allow_invalid_dates.