Step 1: Recall what separates an equation from other expressions.
An expression like $x+1$ or $x-1$ is just a combination of terms, it does not claim anything is equal to anything. An equation, on the other hand, always carries an equal sign $=$ between two expressions. An inequality carries a sign like $\gt$ or $\lt$ instead.
Step 2: Scan through the four options with this test.
$x+1$ has no relation sign at all, so it is an expression. $x-1$ is the same story, just an expression. $x+1 \gt 0$ has a greater-than sign, so it is an inequality, not an equation. Only $x - 1 = 0$ carries the equal sign.
Step 3: Conclude.
Since the equal sign is the one thing that makes a statement an equation, the answer is clear. \[ \boxed{x - 1 = 0} \]