What kind of basic circuit design can block current if volage is about a certain level?
If you are talking about overvoltage protection, there are two general approaches, series protection and shunt protection. Shunt protection refers to a circuit which, when in an overvoltage condition, sends enough current from the overvoltage source into ground to limit the voltage at the protected node. Series protection, on the other hand, refers to a circuit which disconnects a load from a source when it detects an overvoltage condition.

Shunt protection is advantageous because it’s cheap. A single zener or TVS diode can implement it well as long as the source impedance is high enough, otherwise the zener diode may have to dissipate a lot of heat “wasting” the excess voltage to ground.

Series protection is more complicated but doesn’t suffer from the same thermal problems as shunt protection. Practically, series protection and shunt protection are often combined, with the shunt protection device bearing a large load for only a moment until the offending source is disconnected. Once this happens, it has the quality you wanted... no current flows until the voltage reaches an allowed level. 

If you are looking to protect a power line, check out this TI app note on an overvoltage protection circuit for automotive electronics’ DC power inputs, Figure 1 in particular: http://www.ti.com/lit/an/snva190b/snva190b.pdf