The most reliable way of finding out the exact composition of your soil is to have a soil test done. A basic soil test will tell you the soil pH and the levels of important plant nutrients such as phosphate, potassium and magnesium. A basic soil test is usually between $10 and $15. The University of Maryland Extension service provides a comprehensive list of soil testing labs for your area:
After you send your samples you should receive your results in a matter of days.
There are a couple of other methods you can use to get a more general idea of the make-up of the soil in your garden. Go into your garden and pick up a handful and squeeze it in your hand. If the soil holds it shape but is hard to break apart then this indicates a high level of clay. If the soil falls apart then it likely has a high level of sand and if it holds it shape but crumbles when you touch it, then you likely have loam. Loam is the best type of soil for most plants as it is well draining but still holds a good amount of moisture.
You can pick up a soil pH testing kit from a local garden center to determine if your soil is alkaline, neutral or acidic. This will have a large bearing on what plants you can grow in your garden. It is possible to make adjustments to soil pH if you find that your soil is not ideal. The addition of lime (an additive for soil which is made of crushed limestone) will make your soil less acidic (increase the pH), whereas the addition of sulfur will make your soil less alkaline (decrease the pH). In most cases, all that is required to balance the soil pH of alkaline soil is the addition of a good ammonium containing fertilizer.