Until mid-1870, copyright registration duties were handled by the local U.S. District Court of the author or proprietor, while the work itself was deposited in various places; in 1870, all copyright activities were consolidated in the Library of Congress. At that point the early records were supposed to be transmitted to the Library of Congress, but, as Rosen writes, "it’s been fairly well-known that a substantial number of records never made it to the Library, and these records have generally been assumed lost." Rosen describes his interest and the process he followed in unearthing many of the pre-1870 records, on his blog, "Mostly IP [Intellectual Property] History."
The bulk of the copyright records that were turned over to the LoC in 1870 are available on microfilm in the LoC Rare Books Reading Room. They have not been digitized, and Rosen estimates they comprise about 300,000 pages.
Update: And coincidentally, this today from the LoC blog, on more early copyright records.