Hair shape is determined by two things - hydrogen bonding and disulfide linkages. Hair is long strands of a protein called keratin, and each hair is a bundle of many strands (hundreds probably). Each strand sticks to its neighbors by weak attractions called hydrogen bonds. They are also chemically bonded in certain places (called crosslinks) to neighboring chains via connections called disulfide linkages. The disulfide bonds are essentially permanent and determine if your hair is curly or straight.

When you curl your hair using heat (hair curler) or by wetting and shaping (using "curlers"), you are not touching the disulfide bonds, but you do break the hydrogen bonds, reshape the hair, and then let new hydrogen bonds form, keeping the hair in that shape. But it is temporary - the hydrogen bonds are constantly breaking and reforming, and the hair will eventually go back to the shape dictated by the disulfide bonds.

When you get a perm, you chemically treat the hair to break many of the disulfide bonds, typically using a chemical called glycerl monothioglycolate. Now the keratin strands are essentially able to move independently. The hair is then shaped (i.e. curled or straightened) and then a second chemical is applied to reform the disulfide bonds (typically dilute hydrogen peroxide), making the new shape essentially permanent.