The first thing to remember is that having a reaction to a mosquito bite -- like itching or minor swelling around the area -- is typical and not necessarily the sign of an allergy, although some kids do have more severe reactions than others. If the reaction is very severe or unbearably itchy, talk to your pediatrician about it. If you're planning to be in an area where your child is likely to be bitten (like on a camping trip), doctors suggest giving an antihistamine (like Zyrtec, which is non-drowsy) beforehand to prevent the problem. You should also slather your child with a bug repellant (ones that contain DEET are safe to use in children over 2 months), but it's best to spray it on your hands first then rub it onto your child's skin. It's also a good idea to pack a hydrocortisone cream to dab on in case your child does get bitten.
The majority of reactions to bug bites stay localized to the area, but some children may react more seriously, and in very rare cases, even go into anaphylactic shock (although this is much more common in response to things like bee stings). Watch for symptoms like trouble swallowing, difficulty breathing, sweating, and dizziness, which usually crop up within 30 minutes of being bitten or stung. If you notice any of these, get medical help right away.