Question

nails that need to be trimmed once a week
nails that need to be trimmed every two weeks
a result of not keeping your hands moisturized
when the corner of a toenail grows into the flesh of the nail bed
lunula nail matrix nail plate hangnails cuticle fingernails free edge ingrown toenail toenails

Explanation:

Nails that need to be trimmed once a week are usually fingernails. Nails that need to be trimmed every two weeks are toenails. Not keeping hands moisturized can lead to hangnails. When the corner of a toenail grows into the flesh of the nail - bed, it is an ingrown toenail.

Answer:

nails that need to be trimmed once a week - fingernails
nails that need to be trimmed every two weeks - toenails
a result of not keeping your hands moisturized - hangnails
when the corner of a toenail grows into the flesh of the nail bed - ingrown toenail