Rabbi Avraham Danzig was born in Danzig, Poland (which today is known as Gdansk). He studied in Prague. It appears that he lived in Vilna for most of his life where he served as a dayan - judge. However, until very late in his life he never accepted money for his religious activities but earned his livelihood as a merchant.
He was the author of several works. The most famous of his works are the Chayei Adam and the Chochmas Adam. Chayei Adam deals with the laws of daily conduct, prayer, Sabbath, and holidays, the laws discussed in the Orech Chaim section of the Shulchan Aruch. Together with the Chayei Adam he published Nishmas Adam, in which he discusses the halachic issues in greater depth. Chochmas Adam discusses the laws of kashrus and other issues discussed in the Yoreh Deah section of the Shulchan Aruch. Together with Chochmas Adam he published Binas Adam, which parallels the Nishmas Adam published with the Chayei Adam.
Both of these works gained very widespread popularity and have become standard sources for halachic study. In fact, throughout Europe groups began to form called Chevros Chayei Adam which were devoted to the study of Chayei Adam. (There were many such chevros - associations. For example, practically every town had a Chevra Shas - devoted to the study of the Talmud.)
While today the Chayei Adam has been somewhat superceded as a general standard by later works, like Kitzur Shulchan Aruch and Mishna Berurah, the Chayei Adam and Chochmas Adam remain important works which are widely studied.
In addition to these two works for which he is best known, Rabbi Danzig also wrote other books. Among these:
Zichru Toras Moshe - an introduction to the laws of Shabbos.
Kitzur Sefer Chareidim - an abridgement of the classic Sefer Chareidim by Rabbi Elazar Ezkari.
Toldos Adam - a commentary on the Passover Hagadah.
He also mentions a work titled Shaarei Tzedek in his introduction to Zichru Toras Moshe, but I have not yet been able to determine what this book is about