A bitcoin is a unique number registered on bitcoin servers. The numbers are derived by computer algorithm, not all numbers are valid bitcoins. It is kind of like assigning prime numbers a trade-able value.
These numbers can be traded from one individual to another, or traded for goods, in any way money can be used.
In order to find valid numbers, you can run a computer algorithm on your computer, and when it finds a valid number, it registers it with the bitcoin server. If you are the first to try to register the number, it becomes yours, and has real value.
The analogy of gold mining is used to explain it. If you dig in the ground and find gold, you have created value. If you run your algorithm and find a valid number, you have also "mined" value.
why is that valuable? what use is it?