Do our fingernails grow under the skin, or do they grow only near the surface, where our nails are visible ?
The nail is a structure that lies on the surface of the skin. It has a few parts that are important to understand what's going on with it.
The most important part of the nail is the matrix. This portion produces the actual hard part of the nail, and it is supplied by blood and nerves that keep the nail growing. The matrix lies under the nail, closer to the body than--proximal to--the body of the nail. The lunula, the white portion of the nail near its origin (it looks like a crescent moon), is a visible portion of the matrix, so you can tell roughly where your nail starts.
The cuticle is the skin enclosing the end of the nail. Part of the cuticle, the eponychium, is what professional manicurists push in on itself to expose as much of the nail as possible, but it is clear there is some nail body below it that goes (not very far) into the cuticle where the rest of the matrix is contained.
Thus, the answer to your question is a bit complex, but it comes down to whether the depth of the matrix in the cuticle is "deep enough" to say it is growing "under the skin" or if it is only "near the surface." The nail does start growing in the matrix, which is slightly under the skin but which is also very near the surface!