Land patents were the original grants of ownership of public lands to settlers. In Morgan County, the earliest grants date from the 1820s. The patents are available online from the General Land Office Records of the Bureau of Land Management. While that site is easily searchable by name or location, it is not easy to build up a picture of a complete county. This site attempts to pull together all the patents for Morgan County and make them accessible by clicking on maps.
You may access these records by clicking on the map, or via the surname index:
Click on the links to the GLO site (which look like this: MO3200__.056) for full information about the land patent, including access to a document image. Note that the names that appear in this site are as they were transcribed by the GLO site; you can check the document image if you suspect a mistranscription.
The parcels of land are described according to the Public Land Survey System. Morgan County is about six survey townships from north to south (townships 40-45 north) and four from east to west (ranges 16-19 west). Each survey township is about 36 square miles, and is divided into 36 sections. Each section can be divided into aliquot parts, which are normally described as corners of corners. Section 16 of each township was set aside as "school land" and disposed of by the state rather than the federal government, so there are no records for this section in any township in the GLO records.
A typical description of a single 40-acre part might go like this: "the northeast corner of the southeast corner of section 7 of township 41-N in range 17-W". As an example of how this site can be used, the parcel in this description was acquired by John May in 1837, and was purchased a few years later by my Humes ancestors; using the charts on this site you can see how the Humes family subsequently acquired the land around May's original claim. If you look May up in the index, you'll see that he acquired land in two sections in Hawcreek township at the same time.