The three numbers, and where each of them appears. Regulation Z's calculation at 12 CFR 1026.18(b) is explicit: the amount financed is arrived at by "determining the principal loan amount or the cash price", adding other amounts financed that are not part of the finance charge, and "subtracting any prepaid finance charge". A prepaid finance charge is defined at 12 CFR 1026.2(a)(23) as any finance charge paid separately before or at consummation, or withheld from the proceeds at any time. So a fee deducted from the advance reduces what you receive without reducing what you repay. On a mortgage the two figures are printed on different documents and described in plain language: the Loan Estimate discloses the "Loan Amount", which 12 CFR 1026.37(b)(1) defines as "the total amount the consumer will borrow, as reflected by the face amount of the note", while the Closing Disclosure's loan calculations table shows the "Amount Financed" with the required description "The loan amount available after paying your upfront finance charge" (12 CFR 1026.38(o)(3)). Published material on personal loans works the arithmetic of what that gap does to the annual percentage rate, and this page does not repeat it.
The third number is the payoff amount, and it is a moving target by construction. A payoff figure is the principal outstanding plus interest accrued to a specified date, plus any charge the contract makes payable on payoff. Because interest accrues daily on most consumer loans, a payoff quote is issued as of a date and is wrong on any other date, which is why sending last month's figure leaves a small balance behind and a small balance can keep an account open. For a loan secured by a dwelling there is a federal right to the number: 12 CFR 1026.36(c)(3) requires the creditor, assignee or servicer to provide "an accurate statement of the total outstanding balance that would be required to pay the consumer's obligation in full as of a specified date", sent within a reasonable time and in no case more than seven business days after a written request, with a longer allowance where the loan is in bankruptcy or foreclosure or is a reverse mortgage. On precomputed consumer credit generally, 15 USC 1615(c) gives a parallel right within five days, and one such statement a year without charge.
Interest accrues on principal outstanding, which is the whole reason the same rate costs less over time. A rate is a price per period applied to a balance, so a shrinking balance produces a shrinking charge even when nothing about the contract changes. The corollary is the one that surprises people: an extra payment applied to principal does not save its own size in interest, it saves every future interest charge that principal would have generated for the remaining term. Published material on amortization sets out the payment split itself, including how the interest and principal portions of a level payment move against each other.
Capitalization is how principal grows. When interest that has accrued is unpaid and the lender adds it to the balance, the amount capitalized becomes principal and thereafter earns interest itself. This is not a penalty and not a fee; it is a change of category, and it is the mechanism behind a balance that is larger after a period of non-payment than it was at origination. It appears most often in student lending, where deferment, forbearance and certain plan changes are capitalization events, and published material on federal student loans and on loan servicing covers those specific triggers. On a mortgage or a car loan the equivalent arises when the scheduled payment does not cover the accruing interest, which is negative amortization.
Where "principal" does not mean this at all. The same word names the person on whose behalf an agent acts, which is the sense in the phrase "principal and agent" and behind the term "principal-agent problem"; it names the face amount of a bond, repaid at maturity; and in retirement-plan and trust vocabulary it is often contrasted with income. None of those is the loan sense, and nothing on this page reaches them.
