On paper, the concept of land ownership sounds very simple- you pay money and in return you're given unfettered access to a predetermined amount of land. But how much of that land do you actually own? Do you own the sky above it? How about the land below it? What about all the animals that may live there; do you own those too? All of these questions and more define what exactly it means to "own" a piece of land. What about the earth below your land? Well, again, this varies because owning land doesn't necessarily mean you own the mineral rights to it. In a nutshell, the owner of the respective mineral rights to a piece of land is entitled to those substances that could potentially be sitting below a given property and it's not uncommon for them to be sold separately to land and property rights. And there have been cases in the past of home-owners finding out that there is a huge deposit of gas under their home that they don't have rights to, and have no right to stop the owner of the mineral rights drilling for. Beyond the loss of money from that gas if you didn't own the mineral rights, this can sometimes kill any property value your land may have previously had.
However, if you own both the land and all the mineral rights to a property and you live in the US, everything below the ground belongs to you, unless you happen to stumble on an Indian burial ground or something, in which case you have to report it.
Other than that, practically everything is fair game, you can drill for oil and gas or even mine, assuming you have the relevant permits, and anything you find is yours to sell or keep. However, if something below the ground of your property extends to the property of someone else, like a pool of oil or vein of gold, they have an equal claim to the portion of the material on their side of property line. This can get messy very quickly since determining exact boundaries underground is difficult and costly, unless you live in the UK, in which case, the Crown has first dibs on all coal, oil, silver, gas and gold found on public or private property.
It should also be noted that in the United States, thanks to compulsory integration, gas companies potentially have the right to drill under your land from an adjacent property if they have leased or been deeded a certain percentage of land surrounding your land, with this percentage varying from state to state.
As for how much of the land below your property you own, there's no real limit enforced by courts and there have been cases of people being prosecuted for trespassing on other people's property for digging even in the thousands of feet below the ground in the search for oil. Of course, in reality there is a practical limit in that if there is a country on the other side of the world from your land, it's probably not one where your country has jurisdiction. Further, there is the matter of the giant layer of magma inbetween. But beyond that, you can generally dig away to your heart's content as long as you're not breaking any environmental laws and have the appropriate permits.
If you happen to live next to water, your rights again, vary greatly depending on the exact situation and it's covered by a totally different set of laws and rules called, riparian rights. If you have a stream or the like running through your land, in Britain you're entitled to fish while they're in the water on your land, but you can't do anything that would impede the flow of water to other people's land since all running water eventually flows into the public ocean. Likewise, if navigable water flows across, through or otherwise near your property, you're not allowed to do anything that would upset its natural course since the ground beneath all navigable water is owned by the state, so the public can use it freely.
If your property borders a large body of water, on the other hand, in general you have the right to access the water and if you so wish, construct a pier or wharf. However, you have no right to stop people using the water for navigational purposes and your rights to the land beneath the water, if any, only extends to a reasonable level of depth which has never been precisely defined. If you happen to own a chunk of non-navigable water, that's generally yours to do with as you please (barring breaking any environmental laws).
As for animals, you cannot claim ownership of any wild animals on your land; you are, however, allowed to hunt them if they're not endangered. Trees, plants and fruit, on the other hand, are yours to do with as you please as long as the plant was originally planted or seeded on your land.
If your neighbour owns any plants or trees that overhang onto, or the roots spread out under, your property, you're within your rights to trim them back (even the roots) as long as doing so wouldn't kill the plant. However, the original owner still has the rights to any cuttings and can request that they be returned after you've cut them off. The same can be said for any fruit that happens to grow on the tree. So, technically, taking fruit from such a branch hanging on your property is theft.
However, any leaves that fall off of the tree are your problem to deal with, unless they cause damage to your property (for example by blocking up your gutters, or in the case of roots, growing into your septic drainage field or damaging your house foundation) in which case, you can ask your neighbour to pay for the damage and if they refuse, you can sue and will likely win.
In short, if you own a piece of land, you may own more than you'd expect, but in a lot of cases, perhaps less of it than you might wish.