Psalm 27 ("Ledavid, hashem ori veyishi") is recited twice daily from the first of Elul until after Sukkot. There are two customs as to when it is recited. According to Nusach Ashkenaz, it is recited after shacharit and after maariv. According to Nusach Sefarad and Sefardi practice, the second recitation is after mincha not maariv.
The logic behind the Ashkenazi custom is straightforward: once during the day, once at night. That's also when we say the shema, and the purpose appears to be the same: it's constantly on your mind, even if you are not literally saying it for 24 hours nonstop. (See from Shabbat Kedusha: ערב ובוקר בכל יום תמיד, פעמיים באהבה שמע אומרים)
The Sefardi custom, that you recite the psalm at shacharit and mincha, is harder to explain. That means reciting it twice during the day, and zero times at night. The ideal of constant repetition is lost. Why then recite it at these particular times?
The obvious association with these two times is that of sacrifices. In the Temple there are two daily sacrifices, in morning and afternoon. As a consequence, nowadays there are two fully obligatory prayers - shacharit and mincha (maariv is at a lower level). The Sefardi custom appears to view Psalm 27 as a prayer to God. This is distinct from the Ashenazi custom, which views the psalm as a declaration about God.
Where do these two perspectives on Psalm 27 - as prayer and declaration - come from? In fact, both perspectives are already present in the psalm itself.
In the middle of the psalm, there is a section which addresses God in the second person. "Hear, Hashem, when I call with my voice, and be gracious to me, and answer me," it begins. This is quite obviously a prayer. Also, at the very end, is a line addressed at a human being. "Hope for Hashem; be strong and courageous of heart, and hope for Hashem." This is clearly some sort of declaration, not prayer.
Having shown that part of the psalm is declaration and part prayer, the question remains what the rest of the psalm consists of. My instinct is that the rest is a kind of declaration. But for the purposes of the custom we were discussing, it doesn't really matter. The Sefardi custom is based on the example of the prayer found in psalm's middle, while the Ashkenazi custom follows the declaration found in other parts of the psalm.
In the gemara's discussion of prayer one finds (Brachot 28b) the following statement: "One who makes his prayer fixed - his prayer does not consist of tahanunim [entreaties, pleas]." This is a strange formulation. Based on the first half of the statement, you would expect a clear value judgment in the second half, something like "his prayer is accepted" or "his prayer is invalid". Hearing "his prayer does not consist of tahanunim" confuses as much as it teaches. We learn here that a prayer cannot be both fixed and consist of tahanunim. If it is one, it cannot be the other. But then, which of the two should it be? Which one is preferable?
Perhaps the answer is that neither is preferable; both are necessary. There is value in an intimate and personal prayer, and there is also value in a clear and considered declaration of intent. But in practice, it's hard to have both at the same time. All year long, the Sefardi tendency to chant prayers aloud as a community seems to emphasize the "declaration" side of prayer, while the Ashkenazi tendency to say everything in a whisper encourages the personal side. With Psalm 27, the respective emphases of the two communities are reversed. It is Sefardim who emphasize the personal aspect, while Ashkenazim read the psalm as if it were a declaration.