Someone who's ailing is sick. You might visit your ailing grandmother in the hospital.
You can use the adjective ailing to describe someone who's very ill, or use it figuratively, to talk about "our ailing economy," or "the ailing school system." The next time you call in sick to work, you might say, "I can't come in today — I'm afraid I'm ailing." Ailing comes from the verb ail, "trouble or afflict," from the Old English eglan, "to trouble, plague, or pain."