A claim refers both to the discovery of a resource deposit and the claim marker dropped automatically by a character when the claim is found.
There are several types of claims, in increasing order of size (i.e. amount of resources that you are likely to get out of them, but also complexity and cost of building/harvesting them) and rarity. Note that size is not a guarantee of output but merely a general indication of likelihood, as even a tiny can yield a pop, and a remarkable could in theory prove disappointing.
Tiny - Very little. A yield of one item is an unsurprising minimum. It's also the easiest to harvest; just open, put one little bit of nothing in it, equip a tiny enumeration scroll (for the same type of resource as the claim) and click on use.
Small - Bigger than tiny, but still in the diminutive range. Smalls are fairly accessible to harvest. Known small claims call for between one and seven coarse frangible threads (cft). Naturally, the amount of cft used and its quality have an impact on the claim's output. Due to changes introduced in the March 2016 update, Small claims are the most important source of high quality raw materials.
Ordinary - Likelier to fill your inventory. Of course, it's also more difficult to mine. Ordinaries require a bundle created out of various, sometimes rare, items, along with some items of the same type the player is attempting to extract.
Remarkable - As the name implies. Get excited, and get ready to take a bit of a hit on the bundle required.
Sizable - How much size is there to a sizable? Well, it's bigger than a remarkable. Lucky you.
...and possibly others not yet discovered.