Swap is a financial instrument which allows two counter-parties to exchange the conditions or cash flows of a financial instrument they both own with each other. So for instance if I took a loan that had a fixed interest rate and you took a loan that had an interest rate dependent on the market rate we could exchange the terms of our contract. As long as the market rate would stay lower than the fixed rate I would receive the difference between two interest rates from you and vice versa (I would get the interest cheaper since we swapped the terms of our loan contracts). But enough of the science.
One of the big problems that made things worse during the financial crisis was the selling of credit default swaps (CDS) by vast and uncontrolled quantities. By selling a CDS you undertake the risk that the underlying credit will default. So if the underlying credit actually would default the one who sold the CDS would pay to the buyer. So it's kind of like insurance. So by not knowing how much CDS-s had been sold on the market nobody had any idea of the leverage that the market had hence also now knowing the risks involved.
After the markets crashed and the world was about to end the US introduced new financial regulation called Dodd-Frank regulation that was intended to not make the same mistakes again. It was proposed that a single company could not sell more than 100 million USD worth of swaps. After two years of lobbying this was raised to 8 billion per company. There you go - we are back where we started considering that there are currently 119 companies eligible for that amount.
The problem here is that any time that the finance industry creates a new kind of exotic instrument that can be sold and made money off of the regulators are unaware of the risks involved. Regulation is usually introduced after things go bad. And after it is introduced the regulators are lobbied and paid off so that regulation wouldn't actually change anything. The world goes on.