Nuclear power plants are always working, except circa one month/year for maintenance. It's mainly because stopping the nuclear reaction and cooling the reactor down is a very long and difficult process (a few weeks). So "turning on" and "turning off" a Nuclear Power Plant (NPP) is not as simple as pushing a button.
The primary circuit (containing the water heated by the nuclear reactions) is always connected to the secondary circuit (which is a classical Rankine cycle), and the steam of the secondary circuit, which is used to make the turbines turn, and therefore the alternator, thus producing electricity.
As far as I know, the processes for stocking the electricity produced by the NPPs are very limited. I know that there are some kind of big batteries, but their capacity is limited.
This is a big problem in my home country (France), where 75% of the electricity is produced by nuclear energy (we are the most "nuclearized" country in the World), and the electrical consumption is - obviously - much smaller in August than in December.
The consequence is that in August, we produce about 70 GW of electrical power and use only 55 GW of it, and therefore we are forced to sell electricity to other countries because we can't stock it.
For the renewable sources: I know that for photovoltaics (PV) systems, there are small systems of batteries. This is much simpler because the power is way smaller than nuclear power plants (about 1-10 kW by PV system). Moreover, the intermittency of photovoltaics (you don't produce electricity at night, while night is the moment when you consume the most) make it really necessary.
For windpower, you can read a bit about CAES (compressed air energy storage), which is a very promising way to stock energy.
And for thermal power plants (fuel, charcoal) and dams, as someone said previously,
the energy is stored before it reaches the consumer - but that storage happens before generation of electricity. Either in the form of fuel stores, or in a hydro reservoir.