What does the coordinate of time really mean? The meaning of the $t$ which appears in space-time intervals or metrics in general relativity. I concluded that $t$ was just a mathematical thing which allows us to label the "space-time manifold" and only proper time τ had a physical meaning too. On Wikipedia I also found that the coordinate time is not a time that could be measured by a clock located at the place that nominally defines the reference frame. What does it really mean?
First, imagine that you're in an inertial reference frame. We can think of this as a sort of lattice throughout space, with a synchronized clock at each lattice point in space. Now imagine two events, $A$ and $B$, which occur at space-time coordinates $(t_A, x_A, y_A, z_A)$ and $(t_B, x_B, y_B, z_B)$. The reference frame has (again, synchronized) clocks at $(x_A, y_A, z_A)$ and $(x_B, y_B, z_B)$. The coordinate time $\Delta t_{AB}$ is the difference in time measured by the clock at $B$ and the clock at $A$, so that $\Delta t_{AB}=t_B-t_A$. If $\Delta t_{AB}=0$, the events are simultaneous in that reference frame. However, an observer in a different inertial frame (say, in one with a boost $\vec{\beta}$ in the $x$-direction) might measure a different $\Delta t_{AB}$. This means that coordinate time is frame-dependent - it depends on the reference frame you're in.

Proper time is sometimes called "wristwatch time". Now, there's a single clock, and it measures the time between two events $A$ and $B$ on a single worldline. This proper time $\tau_{AB}$ does not depend on your reference frame, but it does depend on the path you take from $A$ to $B$ - that is, it's path-dependent.

One way to think of the difference is that coordinate time tells you something about one reference frame relative to another (because it's frame-dependent), while proper time tells you something about the worldline of the observer (because it's path dependent).