In General, There are two basic types of pain: acute and chronic.

Acute Pain - The normal, predicted physiological response to an adverse chemical, thermal, or mechanical stimulus associated with surgery, trauma, and acute illness. It is generally time-limited and is responsive to Opioid therapy, among other therapies. eg, you hurt yourself with a knife, or by a stove;

Chronic Pain - A pain state which is persistent and in which the cause of the pain cannot always be removed or is difficult to treat. Chronic Pain may be associated with  a long term incurable or intractable medical condition or disease.

By specific pain receptors, pain can divided into Nociceptive Pain and Non Nociceptive Pain

If you according to the pain stimulus, duration, and features, there are also several types of pains:

• Gnawing pain. Continuous with constant intensity. It generally worsens with movement.
  
• Throbbing pain. This is typical of migraine pain. It is caused by dilation and constriction of the cerebral blood vessels.
  
• Stabbing pain. Intense and severe. It is caused by mechanical stimuli.
  
• Burning pain. A constant, burning feeling, like, for example, the type of pain caused by heartburn.
  
• Pressing pain. Caused by constriction of the blood vessels or muscles.