What you are describing is most likely hyperhidrosis, a disorder that causes excessive sweating. It This sweating usually occurs on the underarms (axillary) or the palms of the hands and soles of the feet (palmoplantar).
There is also another type of classification for hyperhidrosis. Primary hyperhidrosis (more common) is excessive sweating not related to another medical condition or medication, while secondary hyperhidrosis is related to a medication or medical condition. Some other differences are that sweating from primary hyperhidrosis does not typically occur while sleeping and that primary hyperhidrosis onsets earlier than secondary hyperhidrosis.1 If you don't have another medical condition or take any medications that may cause the sweating, then the condition you most likely have is primary palmoplantar hyperhidrosis.
The cause of primary hyperhidrosis is mostly related to your genetics, rather than any external causes. Overactive sweat glands are the most common cause and hyperhidrosis is known to run in the family. What actually triggers the sweating to occur is much harder to pin down. Heat can be a trigger, as well as anxiety, but the sweating can occur randomly at nearly all times.
Besides sweating, there are some other possible side effects of hyperhidrosis. One study2 concluded that patients with hyperhidrosis were at an increased risk of skin infection. This can be dealt with by treating your hyperhidrosis, though.
Treating primary hyperhidrosis can be as easy as using prescription or even over-the-counter antipersperants. If those don't work, iontophoresis has been known to help, especially with palmoplantar hyperhidrosis, which would be especially useful in your case. In incredibly drastic circumstances, surgery is available as a last resort. This surgery would involve removing some of the sweat glands. The surgeries have been known to be safe and effective.3, 4
1: Clinical differentiation of primary from secondary hyperhidrosis.
2: Primary hyperhidrosis increases the risk of cutaneous infection: a case-control study of 387 patients.
3: Endoscopic transthoracic sympathectomy: an efficient and safe method for the treatment of hyperhidrosis.
4: Hyperhidrosis treated by thoracoscopic sympathicotomy.
MedlinePlus - Excessive sweating - overview
WebMD - Excessive Sweating (Hyperhidrosis)
Medscape - Hyperhidrosis