What is DDT?
DDT (1,1,1-trichloro-2,2-bis(p-chlorophenyl)ethane) is a white, crystalline solid with no taste and little odor. DDT is an artificial substance; it does not occur naturally in the environment. Commercial grade DDT is rarely pure, and typically includes three closely related forms. Some production processes also created two other closely related chemicals, DDE and DDD. 
Structural diagram: National Institutes of Health
Uses of DDT
DDT was (and in some countries, still is) used to kill pests, including insects on agricultural crops, mosquitoes that carry malaria, and lice that carry typhus. DDD saw occasional use as a pesticide, and a form of DDD was used to treat adrenal gland cancer. 
DDT in the Environment
Even though use of DDT in this country has essentially ended, it is still used in some other countries and therefore continues to enter the global environment.
Most DDT was applied to crops or to the soil. Some DDT was also sprayed on surface water, or ended up in surface water when carried there on soil particles. Still more DDT was sprayed onto crops from the air, or entered the air via evaporation. 
DDT lasts a long time in soil - perhaps decades, depending on soil type and conditions. Breakdown of DDT into other chemicals may occur because of microorganisms in the soil. It may also evaporate from the soil, or be broken down by sunlight. DDT may also be absorbed by plants, animals that live in the soil, and people and animals who eat crops that contain DDT. 
DDT in the air does not last long - generally a few days at most. DDT in surface water may bind to sediments and last a long time at the bottom of water bodies. Aquatic animals, particularly fish that eat smaller organisms, can concentrate DDT and end up with DDT concentrations in their bodies that are much greater than those found in their environment. 
Exposure to DDT
Exposure to DDT occurs primarily through consumption of food that contains traces of DDT, though the amount of DDT in crops - at least those grown in this country - has decreased with time. 
Health Effects of DDT
The NIOSH Pocket Guide to Chemical Hazards  lists the following symptoms related to DDT exposure:
Eye and skin irritation
Paresthesia of the tongue, lips, face
Anxiety, dizziness, lassitude, convulsions
Paresis of the hands
Potential occupational carcinogen
When checked on 7 December 2015, U.S. EPA's Integrated Risk Information System  stated that DDT is a "probable human carcinogen - based on sufficient evidence of carcinogenicity in animals", specifically "observation of tumors (generally of the liver) in seven studies in various mouse strains and three studies in rats." When checked on 7 April 2015, the International Agency for Research on Cancer  classified DDT as possibly carcinogenic in humans.
 NIOSH Pocket Guide to Chemical Hazards, published by the National Institute for Occupational Safety and Health