The typewriter symbolizes the journey of a life; it is linear, words following words in a specific order, typed by a person who wishes to remember, or to be remembered. When a person is typing, it is as if their entire lives can be spelled out on the pages, not lived and then stopped in an instant, as was the life of Oskar's father. In this way, the typewriter travels from a time of great tragedy -- the bombing of Dresden -- through lives and then into a time of terror -- the World Trade Center attack of 2001. As it passes through the hands of Oskar's grandfather, his grandmother, and his own mother, it represents to each a method of finding order in the chaos.
When the pages are in the typewriter, I can't see his face. In that way I am choosing you over him. I don't need to see him. I don't need to know if he is looking up at me. It's not even that I trust him not to leave. I know this won't last. I'd rather be me than him. The words are coming so easily.
(Foer, Extremely Loud and Incredibly Close, Google Books)
One example is Oskar's grandmother, who claims to be writing her life's story on the typewriter, but since it has no ink ribbon, she is turning out blank pages. She also types long stretches with just the space bar, meaning that no key ever hits the paper. This represents both the loss she feels in the disasters, and her feeling that her own life is simply blank spaces of time until her inevitable death. In that manner, the typewriter is a symbol both for life and for the end of life.