Amortization spreads a cost across the periods it relates to instead of charging it in one.
Two uses of the word
Amortizing an intangible asset. Something bought that has value over years but cannot be touched — a licence, a customer list, goodwill from an acquisition — is charged a portion at a time across its useful life. This is the same idea as depreciation, which applies to physical assets.
Amortizing a loan. A loan repayment is part interest and part principal, and the mix shifts over the term. Early payments are mostly interest; later ones are mostly principal. Only the interest is a cost; the principal is repayment of money already received. See debt schedule.
Why it matters to the numbers
Both uses have the same effect: a payment that leaves the bank account is not the same as a cost in the period. This is the distinction at the centre of accrual accounting, and it is why profit and the bank balance move differently.