There are three main water parameters: pH, Hardness, and Alkalinity.
pH is the measure of hydrogen ions in the water. But that is hard to comprehend in reality, so you can think of it as a scale of acidity.
Hardness is the measure of dissolved mineral ions in the water (primarily Calcium and Magnesium). Hard water has a lot of dissolved minerals. Soft water has few dissolved minerals. Hardness is totally independent of pH. You can have soft water at a high pH, soft water at a low pH, hard water at a high pH, or hard water at a low pH. There is no direct connection between hardness and pH.
Alkalinity is the measure of how well the water can resist a change to become more acidic. In other words, how well can the water resist a drop in pH. Some people will say it is the ability to resist a change in pH, but Alkalinity will specifically resist a drop in pH. The reason it works is that when a solution becomes more acidic it does so through an increase in the number of hydrogen ions. The presence of Alkalinity in the water picks up the free hydrogen atoms so they do not make the solution more acidic, or drop the pH. Alkalinity can also be called carbonate hardness, because it is usually carbonates, or bicarbonates, that provide the Alkalinity (the carbonates are the molecules eating up the free hydrogen ions). However, the more general term for Alkalinity is actually a Buffer. A Buffer will prevent a change in pH, while Alkalinity will prevent a drop in pH. You could also have a buffer that prevents a rise in pH.
In reality Alkalinity and pH should be thought of as separate things. In some cases, depending on what type of buffer you use, increasing the alkalinity will increase the pH. But each buffer will behave differently, and the correlation is not always true. For instance if you use baking soda (sodium bicarbonate) as a buffer, and add it to water, the pH will increase, but will max out at 8.3. So if the pH is 8.3 and you add more baking soda, you are still increasing the Alkalinity, but the pH is not increasing. So in general you should think of them as two separate things.
I would also go to this thread, and read the articles provided by Kaiser. They are more in depth than the info I can provide.