When I read some papers about MHD dynamo theory, there is always a $\tau$ that means convection overturning time. What does convection overturn mean?
The convective overturn time is the typical timescale for a convective cell to rise in a gas. Imagine a "lava lamp" - it's the time for one of the blobs to rise from its lowest to highest point.
I am most familiar with its use in stars, where convective energy transport is modelled using a mixing length. This posits that the typical height travelled by an adiabatic convection cell is some multiple $\alpha$ of the pressure scale height $H_p$ (the typical length on which the internal pressure changes significantly).
The convective overturn (or turnover) time is then $\alpha H_p/v$, where $v$ is the velocity at which convective cells rise.