Zero squared is still equal to zero because zero times any real number is zero. Squaring a number simply means to multiply it by itself, or to raise it to the exponent of two. For example, zero squared can be written as 0*0 or 02.
Exponents are a convenient way to simplify large chains of multiplication of a single number. They are small numbers written over a base number. This is equivalent to stating that the base number is multiplied by itself as many times as the exponent is in size. For example, 53 is equal to 5*5*5, which is 125. When a number has a large exponent, it saves time and space to write it as an exponent rather than expanding it.