The Torah (Shemot 13:16) describes the Tefillin Shel Yad (the Tefillin worn on the arm) as an "Ot Al Yadecha" – "sign on your arm." The word "Yadecha" ("your arm") appears in this verse with an unusual spelling, ending with the letters "Kaf-Heh," rather than ending with a "Kaf Sofit," as the word normally does. This led the Sages to read the word "Yadecha" in this verse as if it were written, "Yad Keha" – "weak arm." Meaning, the Torah here alludes to the fact that the Tefillin Shel Yad should be worn on the weaker arm, which, in the case of most people, is the left arm. Left-handed people, however, must wear the Tefillin Shel Yad on their right arm, which is their weaker arm.
This Halacha gives rise to the question of how precisely we define "left-handed" in this regard. It is not at all uncommon to find people who write with one hand but perform other activities with the other. On which hand should they place their Tefillin Shel Yad?
The Shulchan Aruch cites two views on this matter, as to whether we determine one's status based on the hand he uses for writing, or the hand he uses for other activities. The Shulchan Aruch sides with the position that we determine a person's status based on the hand he uses for writing. Therefore, one who writes with his right hand and performs other activities with his left hand would wear Tefillin on his left arm; since he writes with his right hand, Halacha considers him right-handed with respect to wearing Tefillin. Conversely, a person who writes with his left hand and performs other activities with his right hand has the status of a left-handed individual, and must therefore wear Tefillin on his right arm.
However, since this issue is, after all, subject to debate, Chacham Ovadia Yosef ruled that a person in this situation should endeavor to satisfy all views by wearing Tefillin on both arms. Meaning, he first puts his Tefillin on the arm used for other activities, with a Beracha, in accordance with the ruling of the Shulchan Aruch. Later, to satisfy the other position, he should put Tefillin without a Beracha on the arm he uses for writing. Since Tefillin constitutes a Biblical obligation, and an obligation afforded particular importance and severity, one should make a point to perform this Mitzva in accordance with all views.
If a person is completely ambidextrous, and performs all activities with both hands with equal efficiency, then, as the Shulchan Aruch writes, Halacha considers him right-handed, and he therefore wears Tefillin on his left arm.
Summary: A right-handed person wears the Tefillin Shel Yad on his left arm, whereas a left-handed person wears the Tefillin Shel Yad on his right arm. One who writes with one hand and performs other activities with the other hand should initially wear his Tefillin Shel Yad with a Beracha on the arm used for other activities, and then put Tefillin without a Beracha on the arm used for writing. Someone who is completely ambidextrous wears his Tefillin Shel Yad on his left arm.