Corner of my toenail went into my skin, got swollen around where the nail was in. Took the nail out but the swollen is still there. Water/pus comes out?

See below. Soak the toe and see your doctor as you may need and antibiotic for a paronychia.
See a podiatrist. You might make the problem worse trying to dig them out yourself in terms of increased pain or infection. A podiatrist can give you fast pain relief. If this is a recurrent issue, there are permanent procedures that keep this from bothering you again in the future.
Infection. You may have missed some nail in the corner. Ingrown toenails can be temporarily remedied through soaking your foot in warm water, removal of embedded debris from the nail borders, and applying an antibiotic ointment to temporarily soften the corners. If you have redness and drainage, see a professional as you may need to have the ingrown nail border removed and you may need antibiotic.
Soak in warm. Water and epsom salt and apply topical antibiotic. Do not place bandais to tight. Should start to resolve if you to the piece out, if not you need to have a podiatrist remove more nail and or an oral antibiotic may speed up the healing.
Ingrown toenail. Probably additional toenail needs to be removed. The "water/pus" is typically seen with ingrown toenail. See a specialist such as a podiatrist.