Without the Sun, our planet would not have a climate, so the Sun is a natural suspect for any observed climate change. On the longest time scales, the central puzzle is that the Sun does not seem to have had as much effect as we would expect. We have good reason to believe that the Sun of several billion years ago was about 30% fainter than it is now, which, other things being equal, would have frozen the watery planet solid. Mostly, though, the planet of the past was even warmer than the present. The most plausible explanations are that greenhouse gases, especially CO2, made the difference - but CO2 can't be the whole story for the earliest couple of giga-years.
The slow warming of the Sun is a consequence of the accumulation of helium "ash" in the core, and it's a process measured in hundreds of millions of years, but what about shorter term variations? The one we know about is the magnetic cycle of the Sun. The solar dynamo is a phenomenon of the convecting region of the Sun outside the fusion core, and has approximately 11 year cycles of reversals superposed on longer scale variations. The 11 year cycles have a small effect on the total solar radiation received from the Sun, about 0.1 %. This number is too small to produce much direct effect on Earth's climate, but there is considerable evidence of a real effect, not only of the 11 year cycle, but of somewhat larger longer term cycles of so-called Grand Solar Minima.
Interestingly enough, these effects appear to be more regional than global, and this is the number one clue as to their yet to be fully understood mechanism. The solar cycle has a very small effect on total solar insolation, but a considerably larger effect on UV emission. Even though this represents a small percentage of total solar radiation, it has a significant effect on the stratosphere where it is absorbed. One of the most plausible ideas for explanation of the solar cycle effect, justified by some climate models, is that these changes in the stratosphere affect the propagation of atmospheric waves with consequences for the lower atmosphere. In addition, the solar cycle is correlated with emission of energetic particles from the Sun, which affect the polar stratosphere, and anti-correlated with the galactic cosmic ray flux, which have their own conceivable side effect via the formation of cloud condensation nuclei and the global electrical circuit.