The question is what happens in the chair and how much of the nail goes. The published description is short, so it is worth reading word for word.
Numbing first, then the cut
The toe is numbed first, and then the ingrown edge of the nail is cut away. MedlinePlus says your provider numbed your toe with a local anesthesia before the procedure started, and then cut the part of the nail that grew into the skin of the toe.
What that wording covers
The description is narrow. It covers the numbing and the cut, and it stops there. How wide the cut is, what happens to the toe afterward and when it should feel normal are all questions the person doing the procedure has to answer. Ask them before you sit down, not after.
Readers who want the aftercare side can read the MedlinePlus patient instructions for ingrown toenail removal in advance. For the condition itself, there is the MedlinePlus encyclopedia entry on ingrown toenails, Cleveland Clinic's guide to ingrown toenails and the NHS page on ingrown toenail.
Questions to settle before the visit
Ask the foot doctor which part of the nail will be cut and what the office expects of you once you are home. Those two answers cover most of what the short description leaves open. The same questions apply at any office, whether it sits in Itasca or in a nearby village such as Wood Dale or Roselle.
The phrase that sets the scope
The sentence says "the part of the nail that grew into the skin of the toe." It does not say the whole nail. Treat that phrase as the stated scope of the procedure, and ask the person doing it to confirm the same scope in plain words.