How can a sample that is in two different states be at a uniform temperature? In my chemistry course, the professor has discussed how a sample that contains both solid and liquid water would be at 0 degrees Celsius no matter what the proportion of solid to liquid water is. I find this concept to be confusing, because wouldn't the areas of the sample that had already changed from solid to liquid begin increasing in temperature even while the other parts of the sample continue to be in the process of fusion? 

The other part of this idea that confuses me is that if the sample was at a uniform temperature, then why do bodies of water (ex. lakes that freeze in the wintertime) continue to have different temperatures of water within them?
For a system in equilibrium: Think of it as a closed system, with a certain amount of energy shared between the solid and liquid components (discounting the container).  At any instant, some of the faster-moving molecules of the liquid could hit the ice, melting a bit, and thereby losing some of their own energy and freezing.

It takes ~80 cal/g to change ice at 0 °C to water at 0 °C. This enthalpy of fusion, aka latent heat of fusion , "absorbs" any additional heat applied until all the ice has melted. So a gram of water at 0 °C has ~80 calories more heat than an equal mass of ice at 0 °C. If you have a 1 gram chunk of ice and apply 40 calories as heat, then about half will melt.

BTW, pure water does not necessarily freeze at 0 °C... it may supercool, which will change the calculations a bit. See this video or this one for a supercool experiment!

As @MaxW states, in real life most systems are not in equilibrium (they need to be shaken or stirred, despite Ian Fleming's direction). To view a clearly non-equilibrium demo, where there is water in three states at once, see Red Hot Nickel Ball on Ice.