A mana curve is just a chart of how many cards in your deck cost 1 mana, 2 mana, 3 mana, and so on. Plot it out, and a healthy deck usually looks like a hill — fewer cheap cards, a peak somewhere in the middle, and fewer expensive cards tapering off at the top.
The reason it matters: you gain exactly one land drop per turn. A deck with too many expensive cards will sit doing nothing early, and a deck with nothing expensive runs out of gas late. The curve is how you make sure you have something worth doing on every turn of the game, not just some of them.
A deck full of powerful 6-mana spells doesn't feel powerful if you're dead on turn 5 because you never got to cast anything.
— the practical argument for curve discipline
"Curve" itself is a strategy term, not a rules term — but it's built entirely on "mana value," which is defined in rule 202.3.
Comprehensive Rules · Rule 202.3
"The mana value of an object is a number equal to the total amount of mana in its mana cost, regardless of color." Example: A mana cost of {3}{U}{U} translates to a mana value of 5.
"Curve" simply means plotting your deck's cards by mana value on the x-axis and count on the y-axis. Nothing about it is enforced by the rules — it's purely a deckbuilding heuristic built on top of a defined rules term.
How to take advantage of it
Count your curve before you count anything else. When evaluating a deck list, tallying mana values by count is one of the fastest ways to spot a structural problem before you've played a single game.
Ramp effectively lowers your curve. A ramp spell doesn't just accelerate you — it means your "expensive" cards effectively cost less in terms of turns, letting a deck run a slightly higher curve than it otherwise could.
Commander decks can run a bit higher than 60-card formats. With 40 starting life and a slower pace, Commander decks can support more top-end than a typical Standard deck — but the same principle holds: even a Commander deck wants plenty of playable turn 2-3 plays.
Curve considerations change by role. A control deck can run a higher curve since it's not trying to win fast; an aggressive deck wants its curve pushed as low as possible to apply pressure before opponents stabilize.