Anna is Oskar's grandmother. Oskar begins his family saga with the story of how his mother was conceived by an escaped arsonist, Joseph Koljaiczek, whom Anna hid from the police under her skirts. This sets up the reader for what we now know will be a very strange family story. Anna appears as a loving and compassionate grandmother, and Oskar himself feels protected when hiding under her skirts.
Anna is an ethnic Kashubian, one of a Slavic people that settled in Danzig before Poland even existed as a state, so she represents stability in a region that's been changing hands for centuries. She's proud of her heritage. Anna is Oskar's link to the past, and in many ways, she's his model of what a good human being should be: wise, practical, competent, and accommodating. He says:
To this day, I wish I could lie like a toasty warm brick constantly being exchanged for myself under my grandmother's skirts." (10.8)
Anna is left behind when Oskar, Maria, and Kurt move to West Germany after the War.