I guess it's obvious you didn't have an allergic reaction / anaphylaxis so that's good. The stinger shouldn't break off like a bee's, so don't be worried if you don't see it. You can treat the swelling with cold such as an ice pack (don't leave it on for more than 15 minutes at a time) and just make sure to keep the wound clean and dry to prevent infection. Hydrocortisone cream or calamine lotion, if you have it, should help with skin itching/irritation. Antihistamines can also reduce itching. Pain medications such as ibuprofen can help reduce the pain. It should go away by itself within a week.
What were you doing when the wasp stung you?