Roughly:
Sovereign Country – Most “countries” in Natural Earth control their own territory and affairs, don’t have far flung territorial holdings, and don’t have disputes with their neighbors. But things get complicated so there are a few other categories.
Country – Several semi-independent countries can form a single sovereign country (one is more dominant than the others). Example: Denmark and Greenland; also Netherlands, Aruba, Curaçao, and Sint Maarten. See dependency below; where more powers have been devolved to the former dependency.
GeoUnit – Used for map labeling when a country has non-contiguous territories or 2+ top-level regions (England, Scotland, Wales, Northern Ireland in United Kingdom).
GeoSubUnit – Either used in similar way as GeoUnit when the country’s situation is complicated, or as a subdivision of GeoUnits. Example: USA with Alaska and Hawaii and Italy with Sardinia, Isole Pelagie, Pantelleria, and Sicily.
Dependency – Relics of the age of exploration and empire where the sovereign country still maintains presence. Examples: Norfolk Island, New Caledonia, Bermuda, and Dhekelia Sovereign Base Area.
Breakaway – Where the breakaway territory is nominally self-administering. Examples: Abkhazia and South Ossetia in Georgia.
Disputed – Two or more “countries” dispute who’s sovereignty reigns in a territory. Natural Earth uses de facto status to indicate who administers the area and who claims it (see note_brk). Example: Abu Musa Island (Admin. by Iran; Claimed by UAE).
Geo core – The part of a country (separate from the breakaway or disputed area).
Indeterminate – Unloved, disputed, neutral or marginal. Examples: Antarctica, Siachen Glacier, Somaliland, Cyprus No Mans Area.
Lease – The host country “leases” part of their territory to another for a period of time. Payment is not always exchanged or accepted. Example: Baykonur Cosmodrome in Kazakhstan.
Overlay – When there is no formal lease but one country operates in another country’s territory or where part of international peace keeping operations. Example: Diego Garcia Naval Support Facility in British Indian Ocean Territory; Korean Demilitarized Zone (south and north), Korean islands under UN jurisdiction.
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