Mineral claims give someone the right to explore for and/or extract minerals on public lands. Claiming minerals is allowed under the Mining Law of 1872, and is only common for metallic deposits or industrial minerals (i.e., not coal or other fossil fuels). To maintain a claim, the individual must perform a minimum amount of work on the land each year. If the deposit is mined, there are generally NO royalties paid on the resource produced (although taxes are of course paid.)
Mineral patents can be sought if the mineral deposit on a claim is proven to be profitable. While this is becoming increasingly less common, it is important to note that when a claim is patented, the government essentially transfers the land ownership to the individual (or company). This provides sole access to the minerals on the land, as well as the surface resources.
Mineral claims and patents do not generally apply to private lands. Instead, mineral rights must be either purchased or leased to mine on private lands.