Some states are perfectly aligned with a particular measurement. If the system is already in one of these special states (called an eigenstate), the measurement gives a definite answer every time. The value you get is called the eigenvalue. If the system is not in an eigenstate, the measurement gives a random result drawn from the possible eigenvalues, with probabilities determined by how much the state overlaps with each eigenstate.

Imagine shining a flashlight straight at a wall. If the beam is already perpendicular to the wall, it hits cleanly -- that is like an eigenstate. If the beam is at an angle, you have to decompose it into perpendicular and parallel components. An eigenstate is a state already aligned with the quantity you are measuring.

If $|a⟩$ is an eigenstate of the operator $\hat{A}$ with eigenvalue $a$, then measuring the observable associated with $\hat{A}$ returns the value $a$ with certainty. When the system is in a general state $|\psi ⟩$, you can write it as a combination of eigenstates. The probability of getting eigenvalue $a$ is the squared magnitude of the coefficient in front of $|a⟩$. In circuit language, gates are unitary operators (reversible transformations), while measurements are associated with observable operators.