Bitcoin is multiple things:
A new currency, like the dollar or euro.
A payment system to pay with this currency. Just like there's Paypal or credit cards or bank transactions to transfer dollars or euros, there's the Bitcoin software and network to transfer bitcoins.
An innovative technology that allows people to gain consensus without depending on a central party, and without requiring trust.
The future of money. Or actually, the future of any form of asset that can be represented by information, such as: money, voting rights, stock / company shares, land or real estate ownership, temporary rental car access, you name it.
Note that all these examples are already digital today, except now they're recorded and administered by some central authority (e.g. banks, notary services, government registers, etc). Bitcoin technology allows for a safe, decentralized, efficient, free alternative that no longer depends on central authorities.
You could also see Bitcoin as 'digital cash' or 'gold 2.0'.
To understand the basic workings, consider Bitcoin to be a global, public ledger. There's only transactions in this ledger, that transfer amounts of imaginary units from one account to another. This is the same as current banking, except with Bitcoin, there is no bank, company, authority, administrator, or other central party that manages the ledger.
By the way Bitcoin is set up, this ledger is stored, maintained, verified and distributed throughout all bitcoin users over the planet. Therefore when using Bitcoin, nobody can tamper with the database, we don't have to trust anyone, we don't depend on anyone, and nobody can enforce arbitrary rules or impose restrictions. Its security and trust is purely based on cryptography and transparency, rather than the supposedly benevolent intentions of bankers and authorities.
Finally: a worldwide, free, open, transparent system to transfer value, where we no longer depend on 3rd parties.
TL;DR = Bitcoin can do for money, what the internet and email have done for information and communication. In my opinion, this is one of the greatest technological inventions since the coming of the internet, and I think it will truly revolutionize the world as we know it.