So, OP, as you see here the answer is complicated, mostly because the US is huge and specific items can vary by state and smaller jurisdiction. I've try to use the term 'common' and 'generally' wherever possible as there are differences in terminology and application throughout the US.

Buckle up. ;)

TL;dr

There is a concept on the US of a Bundle of Rights in relation to property ownership. When we refer to an ownership type i it is generally Fee Simple or Leasehold. (Yes, there are other types of ownership, but for the purposes of OPs question leasehold and fee simple are most common.)

Fee simple is the land and everything attached to the land, houses, barns, wells everything. Leasehold typically gives you ownership of the improvement (see above), and a long-term land lease. That lease will give you rights regarding what you can do on the land.

Source - I'm a general appraiser (which means I can appraise all property types).

The most common purchase in the US is called Fee Simple. This type generally affords you all of the rights to that property and any improvements in the property. All of the rights include everything above and below within an area limit - air, minerals etc. the improvements often include a home, and other strutual items like a shed or barn or guest home or pool or whatever.

Fee Simple right are subject to government things. For instance, many/most areas have land use conventions; often called zoning but terminology can be regional. Zoning is agreed upon by the area representation and effects what can be built on the property. An office tower may be precluded in a rural area with agriculture. But the zoning also establishes some of the rights I mentioned. It will often indicate the maximum height of a building, for instance.

The other ownership type is leasehold. This is when the improvement is purchased but the land is on a long-term lease. There is no established term for a land lease but in my experience they tend to vary between 50 and 99 years. There is always a renewal clause and it's always convoluted to include appraisers and lawyers and anybody else who can make money on it. Land Lisa's can apply to homes or businesses. The most common example would be homes built on land reserved for native tribes, or properties we refer to as mobile or manufactured home parks or communities. Tribal leases tend to be long. Mobile home park leases can often be year to year, with the land owned by another private individual or corporation.

There are other effects on land rights that can often vary from region to region or state to state. In some states beachfront access for instance is reserved for the public good, and anybody can walk along any Beach. That's not true in every state. The federal government reserves a variety of rights. Often these rights are related to a treaty or agreement and includes a large area not just a single private property.

There are agreements that follow the property sale. They are listed on what we refer to as a deed for that property. The deed is the 'pedigree' of the property it tells you where the property is who owns what rights to the property and establish his ownership. These agreements can come in the form of homeowners associations, covenants, easements, in a variety of other tools utilized in both commercial and residential real estate. These tools are used to give limited rights to a third party. In the case of a homeowner's association this may be an agreement to pay a regular fee to keep the street clean, but if you visit r/hoasucks you'll see that some homeowners associations are a little excitable. They have thoughts and lots of them. But that property is bound to that homeowner's association by deed. An easement as another example might give a utility company limited rights on a property. Easements are very specifically defined within their description so typically it won't apply to the entire property but a percentage of the property. It may include the right to put power lines under the ground it may include the utility companies right of access to that particular little section. Easements can be innocuous and nobody notices (tunnel easements), or they can be a lot and you can see them from satellite image (sidewalk easements, above ground pipeline easement, utility line easements, driveway easements, etc). It just depends. An easement will also include a list of activities that are either restricted and not allowed, or activities that are absolutely permissible in the easement area by the property owner.

There is also a concept of condominium ownership. We see this most commonly in residential towers, but it also appears in commercial things like offices and even small industrial buildings. In a condo, you own everything within the walls of your unit, then pay to an overarching organization of other owners within your property to service the outer envelope of the property and whatever might be on the land; good examples of this might be the roof, the siding, parking areas, common areas. If you look at the deed for a condo in some areas it will include the square footage of the individual unit as well as a small portion of the land, typically commensurate with the square foot percentage of the unit versus the entire building.

Hopefully this is helpful. :)