Refers to one's own grandfather or someone else's grandfather. The kanji お祖父さん is sometimes used for this meaning.
My grandfather goes for a walk every morning.
あの子はおじいさんに手紙を書いた。
That child wrote a letter to their grandfather.
elderly man; old man
A respectful or affectionate term for an elderly man, not necessarily related. The kanji お爺さん is sometimes used for this meaning.
公園でおじいさんが将棋を指していた。
An old man was playing shogi in the park.
親切なおじいさんが道を教えてくれた。
A kind elderly man gave me directions.
The old man in my neighborhood often greets me.
During summer vacation I went to my grandpa and grandma's house.
おじいさん is the polite, standard way to refer to an elderly man or grandfather, while 爺 is casual and can sound rough or derogatory.
おじいさん is the polite register form used to address or refer to a grandfather or elderly man, whereas じいちゃん is the casual, affectionate family register.