The Drake Equation estimates how many intelligent civilisations in our galaxy might be trying to contact us right now. Frank Drake proposed it in 1961 not to give an answer, but to show how many things we don't know.
The Fermi paradox is the contradiction between the high estimates the Drake Equation can produce — millions of civilisations — and the complete absence of any evidence for them. If intelligent life is common, the galaxy is old enough that at least some civilisations should have had time to spread across it, or at minimum to flood it with radio signals. And yet: silence.
The leading explanations range from sobering to terrifying: perhaps intelligent life is far rarer than we assume; perhaps civilisations routinely destroy themselves before they can broadcast far; perhaps they're out there but not talking in ways we'd recognise; or perhaps the universe is so vast that even a busy galaxy looks empty at any given point. Nobody knows. That's what makes this equation worth thinking about.
The Drake Equation multiplies seven uncertain factors — from the rate of star formation to how long civilisations survive — to estimate how many intelligent species in the Milky Way might be detectable at any moment. The answer ranges from effectively zero to hundreds of millions depending entirely on your assumptions. The Fermi paradox asks why, if any of the optimistic estimates are right, we haven't heard anything yet.