The kilogram is defined by a prototype (the "International Prototype Kilogram", IPK) -- basically, a kilogram is by definition the mass of a metal cylinder sitting in a vault in Paris. People have made a bunch of other metal blocks with almost exactly the same mass (as near as they could get), called "sister copies". To measure a mass extremely accurately in kilograms, you need to get your hands on either the IPK or one of its sister copies, and use it to calibrate your super-accurate balance.
There is an alternative approach which is planned to be used eventually. To measure a kilogram you would need a watt balance, calibrated by using the quantum hall effect and the Josephson effect, and also an accurate measurement of local gravitational acceleration.
The second approach seems more pleasing than the first, but is not [yet] used. Why? (1) The first approach actually allows significantly greater precision and reproducibility. Who would have thought that an old-fashioned balance could measure with so many significant figures, or that a metal cylinder could retain almost exactly the same mass over many years? (Well, it doesn't actually retain exactly the same mass, one of the reasons for an eventual switch.) But that's the case. It is so far impossible to make a watt balance with as many significant-figures of mass measurement precision as the prototype approach. (2) A watt balance is not just something any old lab can make, you need years of effort and millions of dollars. Even carrying a resistor down a hallway can change its resistance by enough to mess up the watt balance accuracy. When the watt balance standard is adopted, I doubt the number of places on earth where masses can be super-accurately measured will be any higher than it is today.
As far as I understand, the switched definition will allow super-precise electrical measurements to be much much easier. That benefit is supposed to outweigh the disadvantage that it becomes impossible to measure masses in kilograms as precisely as before.
I am not an expert, sorry for any mistakes.