Bitcoin is a digital currency. You can pay with it on spot, in online stores and make transfers. Although, Bitcoin is different than ordinary currencies you know. Advanced math (cryptography) secures each transaction in Bitcoin. It guarantees that transactions cannot be forged and the currency cannot be stolen - of course if the Bitcoin owner uses standard safety procedures (eg. doesn't pass his private key to third-parties).
Bitcoin transactions registry is called Blockchain. It consists of all transfers of this cryptocurrency. You can use it to check your transaction status. The receiver cannot decline he received the funds.
Bitcoin allows quick and cheap international transfers. It's because the cryptocurrency is decentralized - it has no issuer, there are no middlemen in transactions. When paying with Bitcoin you don't have to worry about bank fees or delays. That's why Bitcoin can substitute fiat currencies in payments and transfers.
Transfers are very fast - usually, they take max. 15 minutes. Due to that fact more and more stores, restaurants and other service points accept Bitcoin payments.