Alkalinity refers to the pH of the soil. A pH of 7 is neutral. Anything greater than 7 is considered alkaline. Alkaline soils are sometimes referred to as sweet.
There are some plants that prefer an alkaline soil, such as lilacs and clematis. If you have alkaline soil and want to plant something that does not prefer a high pH, you can amend your soil with a sulfur product or even peat moss or evergreen needles, which tend to lower the pH of soil.