Sounds like you've probably split the cuticle or it's gotten chapped. This is ususal during colder weather and it's very easy to fix by using a thick hand lotion several times a day or by purchasing cuticle oil and putting that on daily. It'll sooth the redness and irritation sos that it can heal up.
If you've recently had a manicure done at a shop, clean any nicks and cuts with peroxide (burns like heck) and put an antibiotic cream on the cut to reduce the risk of any infections. Unfortunately, the risk of disease from some nail salons have made the news and one must be very careful when having manicures done with less than sanitary equipment.
You should find that the cuticle and skin starts to feel better in a day or two.