In traditional banking the banks keep a record of how much money you have. They deal with all the security to make sure people can't claim they have more money than they do. Bitcoin is designed to do away with this centralisation. Each customer keeps track of their own balance through owning a unique piece of data that identifies a bitcoin as belonging to their account. It's difficult to fake this point of data to the point of requiring more computing power than is reasonable, eg you'd have to spend years running your computer to generate a "fake" bitcoin which actually belongs to somebody else, and if there's two conflicting claims one will be obviously fake because the real one will have a record of that bitcoin's ownership all the way back to when it was discovered. In addition, there are "undiscovered" bitcoins which anyone is allowed to find and claim as their own, although this process is getting harder and harder (see "years of computing time", above). A bitcoin is essentially just a solution to a mathematical function which is so tricky there are only a limited number that are likely to be discovered, hence rareity, hence value, as long as at least one other person wants one.