An "allergy" is your body's negative reaction to some kind of substance, like pollen, dust, mold, or certain kinds of food. When you have contact with the substance that you are allergic to, you start to feel symptoms like sneezing, headaches, swelling, itchiness, and so on.
"Allergies" (with an -s on the end) usually means pollen allergies.
When you know that you have allergies, you may start to call them "my allergies".
People often exaggerate and say that something is "killing" them when they are uncomfortable. For example:
This heat is killing me!
Man, these shoes are killing me.
My boss is killing me!
You use "___ is killing me" in casual situations, when talking to friends or people you feel comfortable with.