Before and after linking the nodes. I am trying to create a partially transparent cornea for context! The entire sphere keeps turning black when I link the mapping node to the gradient texture node.
If a Texture Coordinate input is not present, the space in which the Gradient > Linear is measured defaults to Generated.
Generated space measures 0-1 along the sides of an object's bounding box, aligned to the object's own X,Y and Z axes. In that space, the minimum X of your sphere's surface maps to 0, and its maximum X maps to 1.
An un-rotated Linear Gradient varies along X of the given space. The output of that node in Generated space is 0 at the left of your sphere, and 1 at the right. The Color Ramp expects a 0-1 input. In this case, that's what it's getting, and mapping that range as determined by the sliders.
When you connected the Texture Coordinate node, you chose Object space for the gradient. That space is measured in Blender Units, from the object's origin, along its axes. The sphere's origin is still at its center, and it's 0.172 units in diameter.
In Object space, the sphere's minimum X is -0.086 ,and its maximum X is +0.086. That's the range the Color Ramp is getting as input. All the negative values are off the left of the ramp. The positive values will be mapped to whatever is between 0 and 0.086 in the ramp. In this case, Black.
To get the same results as before, use the Generated output of the Texture Coordinate node
Since your using the object coordinates, I think it might be caused by not applying the transformation. I presume you changed the scale of the object somewhere.
so the solution is to apply the transformation :D
Select the sphere and hit Ctrl+A and select All transformations.