Do solar panels generate variable current or variable voltage?
Solar cells are a PV junction, basically a diode so they have similar characteristics. The voltage is dependent on the amount of energy received from sunlight and the amount of current drawn, so it is load dependent. 


Source: MPPT tracking


  Many solar panels are watt-rated. The generated power depends on
  lighting conditions, so either the current and/or voltage is variable.
  Which one is it?


This image better answers your question, you can see how drawing more current drops the voltage. At a certain point when the load impedance is equal to the source you will get maximum power transfer (shown in the smaller lines on the graph). The different light levels are represented with different colors (W/m^2 is how many watts of sunlight is striking 1 m^2 of area), and lower light levels generate different amounts of power. The problem is there are three variables voltage, current (which are dependent on the load) and the amount of power received by the cell.

So, you need a circuit that can track the maximum peak power point (MPP Tracking or MPPT) to get the best efficiency from the solar cell. Either that or make sure you aren't drawing too much power from the cell to keep the voltage level high.