Interest-bearing debt is any debt where the lender charges a fee for the right to borrow money. This means that the vast majority of debt is interest bearing. While "interest free" debt does exist, it tends to simply be deferred, not interest free.
The main reason lenders charge interest is because they are profit-making entities. Lending money is an investment, and interest is the return on that investment. In this way, they are just like any other business -- they acquire a product (capital) for which there is low supply and high demand, and sell it for a profit. When a lender lends money, it is actually selling that money at a profit, which manifests itself as interest.
Debt with deferred interest is debt where the borrower is not obligated to pay the interest until a certain period. This means that rather than shrink with each payment, the amount of debt -- and interest, which is a percentage of that debt -- increases.
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Interest-bearing debt with a fixed interest rate has a rate that does not change over the course of the payback period. This means that monthly costs are fixed.
The interest rate on a variable interest-bearing debt is subject to change over the course of the loan. This means it can be lower than a fixed-rate loan but it can also be substantially higher, and monthly payments are not fixed.