Company A issues its fixed-interest bond and Company B issues a floating-rate loan. They then agree to swap their interest payment liabilities...
The way a borrower can most easily raise money isn't always best suited to their needs. Imagine two firms need to raise money. Company A might easily be able to raise fixed-interest money, but what it really needs are floating rate funds. The reverse is true for Company B. The solution for them would be an interest rate swap.
Company A issues its fixed-interest bond and Company B issues a floating-rate loan. They then agree to swap their interest payment liabilities, and so pay one another's interest and end up getting the money in the form they need it at a cheaper rate than if they had borrowed it direct.
In practice, rather than finding a direct counterparty for the swap, companies will arrange it with a bank that will either find the counterparty for them or act as counterparty itself. Currency swaps are where Company A wants dollars but can get better trading terms in euros, while the reverse is true for Company B, so they swap so that each ends up with what it needs. With many swaps, both interest payments and currencies are exchanged.
Watch Tim Bennett's video tutorial: Beginner's guide to investing: What is a swap?