Klarna just made its debut as the biggest IPO this year. The Swedish buy-now-pay-later giant sees America as its future—and it's taking direct aim at credit cards.

Klarna has been attempting to convince the market that it's more of a digital retail bank than a simple buy now, pay later firm.

Klarna priced its long awaited initial public offering at $40 a share late Tuesday, valuing the Swedish buy now, pay later company at more than $15 billion.