user34354: Every time a hash is successfully created, a reward of 25 bitcoins is paid. Who pays this reward, and on what authority? How does this become legal tender? How is it not counterfeiting?

user34132: It is not legal tender, and there is no one that's forced by law to accept it.
It is a digital token with the properties of money and (with the current state of computation) not possible to forge or to double spend. It is not backed up by anyone, but its users.
The users are the ones who give value to it, and that's because bitcoin is useful.
Currently it is a very good way to make quick payments to individuals without bank accounts, can be transferred to anyone, including people that live in countries without support from payment companies (like those that live in countries that cannot accept paypal payments)
It is decentralized, meaning that you can't take it down by attacking only one entity. And there is no central authority that can alter the history of bitcoin transactions.
Instead of thinking of it like fiat money, you should think of it as digital gold, it is scarce, it is expensive to obtain and its value isn't backed by any government, is backed up by people willing to buy it and use it as a currency or a store of value.
Obs: I edited the answer to clarify some points, but the accepted answer in this SE thread could give you a lot more insight on what gives value to it.

34354: <p>Every time a hash is successfully created, a reward of 25 bitcoins is paid. Who pays this reward, and on what authority? How does this become legal tender? How is it not counterfeiting?</p>
 34132: <p>It is not <strong>legal tender</strong>, and there is no one that's forced by law to accept it.</p>

<p>It is a digital token with the properties of money and (with the current state of computation) not possible to forge or to double spend. It is not backed up by anyone, but its users.</p>

<p><em>The users are the ones who give value to it</em>, and that's because bitcoin is useful.</p>

<p>Currently it is a very good way to make quick payments to individuals without bank accounts, can be transferred to anyone, including people that live in countries without support from payment companies (like those that live in countries that cannot accept paypal payments)</p>

<p>It is decentralized, meaning that you can't take it down by attacking only one entity. And there is no central authority that can alter the history of bitcoin transactions.</p>

<p>Instead of thinking of it like <em>fiat money</em>, you should think of it as digital gold, it is scarce, it is expensive to obtain and its value isn't backed by any government, is backed up by people willing to buy it and use it as a currency or a store of value.</p>

<p><em>Obs:</em> I edited the answer to clarify some points, but the accepted answer in <a href="https://bitcoin.stackexchange.com/questions/182/where-do-bitcoins-come-from-and-what-gives-them-their-value#answer-185">this</a> SE thread could give you a lot more insight on what gives value to it.</p>
