It's not clear how you plan to use this diode since "measurement setup for capacitor leakage" doesn't explain anything. However, in general, the "speed" of a diode is it's reverse recovery time.
Power rectifier diodes are generally slow, but are optimized for handling large current and substantial reverse voltage. "Switching" diodes or "high speed signal" diodes are more optimized for speed, like the common 1N4148.
The reverse recovery time of a diode is the time it takes to stop conducting after being switched from forward to reverse biased. This is a issue with semiconductor diodes because any charges in the depletion region will continue to allow conduction even when reverse biased. It takes a finite amount of time to sweep out these charges.
Schottky diodes are generally much faster, usually to the point you can think of them as instantaneous for the purpose of most circuits. This is because they have a much smaller depletion region inherent in their construction. Intead of a N-P junction, they are a semiconductor-metal junction. The flip side is that they also leak more and are unavailable or very expensive above about 100V.
Vacuum tube diodes work on a completely different principle and therefore have their own set of characteristics.