Tefillin are referred to as a sign (“It shall be a sign upon your arm”). Shabbat is also referred to as a sign (“It is a sign between Me and you,”) and the festivals are considered to be a sign as well. For this reason, we only put on Tefillin on weekdays and not on the Sabbath or festivals (Rosh HaShana, Yom Kippur and the Three Pilgrimage Festivals: Pesah, Shavuot and Sukkot).
Regarding putting on Tefillin during Hol HaMoed (the intermediate days of Pesah and Sukkot), there is a divergence of custom. In Eretz Yisrael today, the custom is not to put on Tefillin on Hol HaMoed. Outside of the Land of Israel, Ashkenazim (Lithuanians and German Jews) put on Tefillin on Hol HaMoed; Hasidimand Sepharadim do not.
On weekdays, Tefillin are worn during the day, but not at night. Even though, according to Halacha, Tefillin can be put on throughout the day, we wear Tefillin only during morning prayers. This is because Tefillin require pure thoughts and it is forbidden to divert one’s thoughts from them. There are individuals who are constantly involved in Torah study and can keep their thoughts pure throughout the day. They wear Tefillin even when they are not at prayer.
Whoever cannot wear Tefillin for the entire prayer-service (such as one who is sharing his Tefillin with another), should at least put on Tefillin for Shem’a Yisrael, so as not to “bear false testimony”, i.e.: in Kriat Shem’a we say “You shall bind them as a sign upon your arm and they shall be frontlets between your eyes,” and therefore he should be wearing Tefillin.
The Tefillin Hebron staff is at your disposal regarding any question that involves putting on Tefillin.