One of the many wonderful collections of digitized records available online at FamilySearch is the Bureau of Land Management Tract Books, "United States, Bureau of Land Management Tract Books, 1820-1908." Unless you are already familiar with how to use tract books, trying to use them online can be a bit confusing at first, but a great resource in the free FamilySearch Research Wiki makes the task much easier.
Beginning with the BLM General Land Office's free patent search, I located a homestead patent for Charles P. Ingalls in De Smet, South Dakota (then Dakota Territory). The patent description (the patent image is not available online) indicates that the 160 acres was located in Section 3 of Township 110, Range 56.
Once you have the Township, Range, and Section information from the BLM General Land Office database, the next step is the FamilySearch Research Wiki where a time-saving Tract Books Coverage Table simplifies the process of identifying which volume to search for a particular piece of land. Without this handy tool (thank you to whoever created it!) you would have to open volumes one by one to identify which one covers your area of interest.
Scrolling down the list to Dakota Territory, we can see that Township 110, Range 56 should appear in Vol. 21-23 of Dakota Territory. Based on the coverage of these three volumes, I began with Volume 22. While tract books from different land offices and different time periods might be formatted differently, they are arranged with records appearing in order by the legal description of the land: by township, range, section, etc. In some cases this might mean that all records of one township appear in order by range, and in other cases they might be arranged in a particular volume (containing two or more townships), by range first and then township.
T110, R56 begins on page 109 (image 114) of Dakota Territory Tract Books, Vol. 22. Section 3 appears near the bottom of the page, where we learn that Charles P. Ingalls was able to successfully complete a homestead claim of 160 acres, paying a total of $10.00 in filing fees for the free land. The sale occurred on 27 February 1880, and the final certificate was issued on 11 May 1886.
While tract books are interesting for learning a few additional details beyond the patent entries on the BLM website as in the above example, where they are most helpful is for locating land sales that were cancelled or relinquished prior to the patent being issued, such as failed homestead claims. To search for these entries, use census, tax, and other records to determine the township in which your ancestor lived, and then browse the tract books for the entries from that township. For example, a cancelled homestead for Charles P. Ingalls in Section 4, Township 109, Range 38 appears in the Minnesota Tract Books, pointing to the locust devastation which led the Ingalls family to leave Walnut Grove for Dakota Territory before proving up on the homestead.
Once you have the legal land description of your ancestor's land, whether from the BLM GLO database, or the Tract Books, don't overlook ordering the actual Land Entry Case File! To see what one looks like, view the case file for the DeSmet, S.D. homestead claim of Charles P. Ingalls on the National Archives website.