Bitcoin is money.
The difference between Bitcoin and the currency you are accustomed to using is that the tasks of issuing Bitcoin and keeping track of who has how much of it are performed by mathematics, computer programs, and a dispersed computer network, as opposed to human institutions and laws.
How do you get Bitcoin? You buy it with some other currency, or you can mine it.
We already buy currencies all the time. If you take a trip outside the country, you have to "exchange" the currency you have for the local currency -- but the word "exchange" is misleading because really what you are doing is purchasing it, and the price to purchase that currency in your own can and does fluctuate.
The "mining" process of Bitcoin is novel compared to issuing national currency.
There is a single, unified Bitcoin network comprised a number of servers, and these servers record all Bitcoin transactions. That is one of their two big functions. The second is to facilitate Bitcoin "mining."
Let's say I ask you to figure out whether or not the number 119 is prime. Currently, even the most advanced computer algorithms we have to find the answer to this question basically just do glorified versions of guess-and-check, because there is as yet no way to quickly figure it out.
Now let's say I asked you to figure out 7*17. You could probably do it in your head in under a minute.
This is an example of a mathematical function with asymmetrical difficulty: doing something in one direction is easy, doing it in another is very hard. A different mathematical challenge with asymmetrical difficulty is the basis of bitcoin mining: the servers have a few standing "questions" which require immense computing power to solve, and if you solve them, then the servers verify your solution, and if it is correct, then you get some bitcoin.
About the first function of the servers -- keeping track of transactions. Let's say I have $7000 in the bank, and I write you a check for $500. You deposit it. My bank deducts $500 from my account, and adds $500 to yours. No physical movement of physical tokens is needed. It's all "software."
The same thing is true with Bitcoin, except that instead of two banks keeping track of the transactions, the Bitcoin network itself keeps track of the transactions.
Why would someone want Bitcoin?
(1) It's traded like any other currency. If someone thinks the price of Bitcoin is going to go up, then they could make a profit by buying some.
(2) Bitcoin is stored in Bitcoin Wallet, which unlike a bank account, does not require you to provide a name, address, SSN, or other personally identifiable information. It's like having a numbered account -- this is the "crypto" part of the cryptocurrency.
(3) The overall supply of Bitcoin is (arguably) more stable than the supply of other forms of money.