The Time Value of Money Calculator

Help me with this time-value-of-money calculation and explain the logic.

What I'm trying to find: [PRESENT VALUE / FUTURE VALUE / PAYMENT / RATE /
PERIODS]
Known values: [AMOUNTS, RATE, NUMBER OF PERIODS, TIMING]
Compounding: [ANNUAL / MONTHLY / OTHER]

Provide:
1. The answer, with the formula and each number plugged in
2. A plain explanation of why the money is worth what it is
3. How the answer changes if the rate or time shifts meaningfully
4. The intuition I should carry away for similar problems

Show your working so I can reproduce it.