For a long time many doctors believed that babies under 2 couldn't develop allergies because their immune systems were not developed enough. (Allergies occur when the immune system recognizes substances, like pollen or pet dander, as foreign and launches an attack against them, triggering symptoms.) But now we know that's not true. A baby can have an allergic reaction to almost anything she's exposed to on a consistent basis, including dust, mold, pet dander, or even certain foods.
On the other hand, seasonal allergies (to things like pollen or grass) tend to manifest later because it takes at least one allergy season for a baby's immune system to "learn" to be allergic to them. Depending on when your baby was born, you're not likely to notice seasonal allergies earlier than 12 to 15 months.
The symptoms of allergies in babies and young children include a runny nose with clear mucus, itchy, watery eyes, sneezing, hives, and dark circles under the eyes (called allergic shiners). If you suspect your child has allergies, talk to your pediatrician. If your child is diagnosed with allergies, your doctor may suggest you see an allergy specialist, but most pediatricians can manage mild to moderate allergies very well with over-the-counter antihistamines like Zyrtec (for kids 1 and up) or Claritin (for kids 2 and up).