The two types of identity properties are the Identity Property of Addition and Identity Property of Multiplication.

Identity Property

Official Book Definition: This property says the sum of any number and 0 is equal to the number for addition. The product of any number and one is equal to the number for multiplication.

Example:
Addition... 9,876+0=9,876
Multiplication...6,786x1=6,786

Personal Definitions:
Ayra's definition: For addition, the identity property is when you can add any number to 0 and get the original number. For multiplication, you can get and multiply it by 1 for the same number.
Carleigh's definition: The identity property is when you add any number to 0 and it is still the number.
How to Remember: The way to remember is just the word identity. Every time the identity property is used, the number is identified the same way. The number has the same identity.