Armpits can turn red and itch for many different reasons. There are simple causes, yet some are signs of a serious medical condition that requires medical attention. If your symptoms are not improving in the next few days or if you have other symptoms I would recommend seeing your primary care physician
The most common cause is an allergic reaction. It is possible that you have developed a delayed reaction to the deodorant. Many people develop this, especially those with fragrances. I would recommend switching deodorants.
Another common cause is an infection of the hair follicles. While a frank skin infection ( a cellulitis) is possible, the more common cause is folliculitis (an infection of the hair follicles). This will be warm and red and painful. It is rarely itchy. Topical antibiotics or even pills are needed for this.
A more serious cause is hidradenitis suppurtiva -- an infection of the sweat gland that causes severe scars and can be debilitating.
If you are having fevers and chills I would recommend seeing your doctor
Otherwise, I would recommend changing deodorants. Talcum powder can be used for a few days to let things "cool down."
I would recommend seeing your doctor if symptoms persist. Good luck!