To hospitalize is either to check a patient into a hospital, or to injure someone seriously enough that they need to be treated in a hospital. If you get a bad infection, your doctor may decide to hospitalize you.
Any time doctors decide that a patient ought to be treated intensively or needs a serious kind of surgery, they hospitalize their patient. It's more common to find this verb in the phrase "was hospitalized," as in "My favorite basketball player was hospitalized after collapsing on the court." Grammar snobs have long disapproved of verbs formed with the -ize suffix, and hospitalize was controversial when it was coined around 1870.