My big toenail is white toward the upper left side, and this discoloration extends about halfway down. What might it be?

Fungus among us. This may be onychomycosis, a usually benign fungal infection of the toenail. It is not serious and can be treated with topical or oral antifungals if desired for cosmetic purposes.
Onycholysis. Depending on the states of the nail (thickened, crumbly, painful, or with debris (i.e. Powder) under the nail, it may be just that the toenail is lifting off the nail bed on which it covers. When the nail lifts from the bed it appears to change color because the skin beneath it does not affect the appearance. The other aforementioned symptoms are indicative of fungal infection.