One of the common difficulties people have is in understanding bitcoin is the role that block creation plays on the network and the notion of what a "coin" really is. I think for people who are not crypto geeks like many here on the forums, a much better description of this process could be given by making an analogy to deeds for things like houses. If you think this is a good way of explaining the bitcoin network you are free to use the analogy below any way you want (even copy pasting it), if you would like you can make a donation to the address in my signature. Here is how it would work...
Suppose Alice builds a house in a city somewhere. After this is done she will own the newly built house and to prove she owns it she can point to a deed at city hall that looks something like this: (New House, Alice). The two entries on the deed are the (seller, buyer) of the house and represent signatures at the bottom of the deed. In this case there is no seller since it was a newly built house so Alice just signs as the buyer accepting ownership of her newly built property. Now suppose that at some time later Alice decides to sell the house to Bob. He gives her $100,000 and they go down to city hall to have a new deed drawn up an filed in the vault: the new deed looks like this (Alice, Bob). Since there is now a deed on file that shows Bob as the last person in the chain of owners going back to when the house was built, Bob can now say he owns the house because he, and he alone, has the authority to sign a new deed that looks like (Bob, someone) at any point in the future.
Now suppose that people wanted to start using houses as a currency. Houses are much too valuable to be traded for ordinary day to day purchases so some way of making "change" must be provided. To do this what city hall could do would be to offer the following service: instead of issuing deeds only for the whole property, they would also allow sellers to sell just some portion of their property. Returning to the house Bob owns, suppose he wants to use some of it to buy a $10,000 car from Eve and Eve is willing to accept 10% of Bob's $100,000 house as payment. Bob and Eve go down to city hall and have a new deed drawn up that looks like this (Bob, Eve 10%). This means that there is now not just a single chain of (seller, buyer) deeds going back to the original buyer, but a chain that has a "fork" in it. For this particular house, Eve is the owner of 10% of it since she could decide to have a new deed drawn up transferring all or part of her 10% stake in the house, and Bob owns the other 90% and can have deeds drawn up whenever he wants transferring all or part of that for things he wants. City hall simply keeps filing the new deeds in the vault and also checks every time it allows a new one to be drawn up that the seller really owns the portion of the house they are selling. The way they do this is by simply following the chain backwards starting at the last deed that the seller has on file and making sure it can be traced all the way back to the original builder of the house. If such an unbroken chain of deeds exists than the seller can transfer that share and they let the new deed be drawn up and filed, otherwise they refuse to put it in the vault with the other deeds.
This chain of ownership may get quite long and even fairly complicated with small chunks of the house being broken off and transfered to other people and then later on some of those pieces may be recombined to make larger chunks (for example a car dealership selling lots of cars in exchange for 10% chunks of the house). At any given time there may be hundreds of owners each owning a bit of the house but if you add up the shares of all those people at any point in time they will always add up to exactly 100% of the house.
This process is exactly analogous to the way that the ownership of bitcoins works. New coins are created out of thin air by "miners" which are computers that repeatedly solve a math problem needed to make the bitcoin network secure. This devotion of computing resources to running the network is analogous to Alice building the house originally. The problem that they are solving is basically doing the work that city hall was doing in the above example: they are checking that newly created deeds are valid by tracing them back to the original miner that created them and then storing the valid deeds in a vault every ten minutes. Just like in the example above, ownership of the bitcoins can be transferred from Alice to Bob by adding another deed on the end of the chain of deeds with Alice as the seller and Bob as the buyer. Furthermore, they don't have to transfer their entire stake in bitcoins, but are allowed to transfer portions of it down to very small amounts (0.00000001 of a bitcoin is the smallest "piece" of a bitcoin that the system will allow you to trade).
The reward of 50 new bitcoins being created for each block solution found by a miner will eventually be cut in half to 25 per solution. It will continue to be cut in half every so often until the total number of bitcoins in existence adds up to exactly 21 million. This is like the city above running out of land to build new houses; ownership of existing houses could and would still be traded there just won't be any deeds for new houses showing up, only deeds for transfers parts of already existing houses being passed from person to person. After the supply of new coins dries up the reward for mining will dry up as well. As this happens the network will gradually transition to a system of transaction fees set by people willing to make a transfer of bitcoins (i.e. to have a new deed drawn up, verified, and locked in the vault). The sender of bitcoins will specify who they want them sent to, plus some other amount they choose to be given to whoever finds the block solution that allows the deed to be sealed up making the transfer final. Since the sender of the coins sets the fee themselves (and they can set it to 0 or any small amount they wanted to) the miners will compete to claim these fees. Some miners will work only on trying to very quickly find solutions for transfers with large fees in them, while other miners may choose to harvest the many other small or 0 transaction fee transfers and try to make money that way. By this process a market price for different service classes will evolve, people willing to pay higher fees will get expedited service, while people paying little or no fee will still get through but will have to wait longer for their transfers to become finalized. People not wanting to pay fees but still do transfers quickly would be able to set up their own mining computers (something anyone anywhere can do) and try to find solutions for block that contain their transactions, effectively doing the job of city hall themselves.
Because the supply of bitcoins is finite, divisible, and easily transferable they satisfy the criteria to be used as a currency, just like the houses above could be used as a currency. The bitcoin network is not theoretical, it is up and running now and has been processing dozens of transactions per hour for some time now. If you want to start running the mining software that supports the system (like building new houses) or start trading things to other people for some of their bitcoins (like Bob buying Eve's car) you are free to do so at any time and since the system is open, you will be able to continue trading them as long as you are willing to, even if that means running the network yourself.