My daughter is 16. Has 2 bites like mosquito bites but there's a large red ring around them like a rash. It is itchy. Can anything be done for them or just run its course .

Not knowing what kind of bites they are, it is impossible to say how to treat them. Where was she recently ? Hiking, visiting the beach, what ? Of course if they are itchy, you can try applying dabs of white vinegar. This may control the itching. If you are concerned, take her to the doctor if they are not gone in about a week.

First guess, depending on where you live, is a spider bite, possibly one that needs medical care.
Very few are poisonous. I've had them as a kid that covered half my arm.
2 - Ringworm is a well defined red ring and SOMETIMES has a middle spot. You need a doctor for that (it's not a worm; don't worry).
3 - Not likely from the sound, but tick bite with infection such as Lyme. Usually a round rash without clear edges, much bigger than ringworm or a spider bite, but not necessarily.

You could spend all day looking online at pictures of all those.

I use cortisone cream on all bites, rashes, anything itchy.

I've used a product called After Bite successfully on different types of insect bites. Its available at most drugstores.

One up to AfterBite, many summers with that and calamine.

Common Bites:

What Bit Me? How to Identify Common Bug Bites | Everyday Health (http://www.everydayhealth.com/skin-and-beauty-photos/how-to-identify-common-bug-bites.aspx)

Put your thumb over an open bottle of ammonia and shake the bottle. Then dab the droplet on the bite. Makes the itch go away. Afterbite is 90% ammonia.

Jam3sFord, thanks for helping out, but this post is too old for more suggestions. Try to keep responses to much more current ones.
(Also, she didn't say she knew they were mosquito bites.)