Grandparents are the father or mother of a person's own father or mother, being respectively a grandfather (also colloquially grandpa, grandad, gramps, pop or many other terms) and a grandmother (also grandma, granny, grandmama, grandmums, gran, nana or many other terms).
When used as a noun (i.e., "…a grandparent walked by"), grandfather and grandmother are usually used, although grandpa/grandma and granny are often used. When preceded by "my…" (i.e., "…my grandpa walked by"), all forms are common (anywhere from "…my grandfather…" to "…my gramps…"). All forms can be used in plural, but gramps (plural gramps) is rare.
In writing, grandfather and grandmother are most common. In speech, grandpa and grandma are most common in the US, where grandfather/-mother is very rare when referring to a grandparent in person. In Britain and New England[1] nan, nana, nanny and other variations are often used for grandmother in both writing and speech.
Numerous other variants exist, such as gramp and grandpap or pop for grandfather and grandmom, grandmama and grammy for grandmother, etc. Because of the terms' unavoidable familiarity, there are many simplified versions as well, including gogo,grampy, granddaddy, grandpappy, etc.
Given that people may have two living sets of grandparents, some confusion arises from calling two people "grandpa" or "grandma", so often two of the other terms listed above are used for one set of grandparents. Another common solution is to call grandparents by their first names ("Grandpa George", "Grandma Anne", etc.) or by their family names ("Grandpa Jones", "Grandma Smith"). In America (where most families are of mixed ethnicity), many families call one set of grandparents by their ethnic names (i.e., Hispanic grandparents might be called "Abuelo" and "Abuela", French grandparents might be called "Pépé" and "Mémé", or Dutch and German grandparents might be called "Opa" and "Oma"). As you can see, there are many different types of names for "grandma" and "grandpa" for different languages and cultures.
The parents of a grandparent are called all the same names (grandfather/-mother, grandpa/-ma, granddad/-ma, etc.) with the prefix "great-" added. Thus, one's father's father's father is a great-grandfather. The same applies to one's great-grandparent's parents (great-great-grandparents). Also note that two individuals who share the same great-grandparents but are not siblings or first cousins are called Second Cousins to each other because second cousins are the grandchildren of your grandparent`s siblings.