Discount- Usually it is used in reference to present value of future cash flows. If you expect to receive $110 a year from now, discounted at 10%, the PV of that cash flow would be $100. You discount the $110 using the 10% rate (or whatever rate you decide on based on risk, inflation, risk-free rate, etc.) back to it's value today.
Amoritize- Same concept as depreciation, only it is technically for intangible assets, such as patents, copyrights, etc. When you amoritize a patent, you allocate the expense over the patents life. You buy a drug that is patented (patent expires in X years), the cost of that aquisition is amoritized over X years. The amortization expense per year depends on how the company decides/legally allowed to amoritize the patent.
Expense is just a cost. Interest expense on debt/bond is the same as the cost you pay to borrow that money. You have one year long loan for $100, the interest expense is $10, it cost you $10 to borrow that $100 for the year, on top of repaying the $100.