Change of variable from t to \$\tau\$ during convolution
In addition there should be somewhere mentioned the integration period, it's from time=0 to current time t.

$$\int_0^t{x(\tau)\cdot h(t-\tau)d\tau }$$

Tau is the formal variable for integration process, t is the running physical time. The integral should be calculated completely from blank board for every t. That's quite a job for a signal processor. Fortunately there are  accelerating algorithms developed.