Assuming you mean the equation $\kappa \frac{\partial^2u}{\partial x^2}=C_p\frac{\partial u}{\partial t}$, then they are all meaningful. In that case $u(x,t)$ is the temperature at point $x$ and time $t$.

So $\frac{\partial u }{\partial t}$ represent the variation of temperature at a particular position and time, with respect to time. Thus, it represents a cooling or a heating, depending on its sign. In response to your comment, yes it can be said that it is the rate of change of temperature with respect to time (and not position!).