What is meant by a 'hydrophilic' sphere? So, I know that a hydrophilic material is defined as those with a special affinity for water, opposite to 
hydrophobic, those that naturally repel water.

I am investigating the impact of spheres with water surfaces, and the splashes caused by them. However, one research paper focuses on the difference between the fluid dynamics of hydrophilic spheres and hydrophobic spheres, and the difference in the cavities they make when they sink underwater. 

I understand what a hydrophobic sphere is, but what is a hydrophilic sphere? Like a rolled up ball of tissue paper? Or the water absorbing waterbeads/orbeez that get bigger(the ones children play with)?
The answer to your question depends on the size level (microscopic or not). But it sounds like you are referring to spheres at a size that you can hold, or larger.

A hydrophilic sphere could simply be a sphere of some solid material (like plastic), coated with a layer of hydrophilic material/substance. 

This page on hydrophilic coatings has some information on what this material may be.