The difference is that the board-lasted shoe makes for a more rigid and stable shoe for runners who over- or under-pronate, or who wear orthotics. Slip-lasted shoes are good for those who have rigid feet and need more freedom of motion; it also makes for a lighter shoe. The combination last, of course, provides the benefits of both worlds by providing stability in the heel and flexibility in the forefoot.
Heel Counter. The stiff material at the back of the shoe is built to resist too much motion in the ankle. If you overpronate, you should look for a rigid heel counter. Give it a squeeze to see how firm it is. At the top of the heel counter is usually a chunk of padding called an ankle collar which is intended to protect cushion the ankle and the achilles tendon.
Uppers. This is the portion of the shoe that covers the top of your foot. The uppers of almost every running shoe are made of nylon or nylon mesh, which is lightweight and breathable and, unlike leather, doesn't stretch when wet.
Know your foot
In order to pick the right shoe, you have to know a bit about your foot. Not to fetishize, it's a rare thing to lay eyes on a perfect pair of feet. There are very few runners lucky enough to have feet so biomechanically efficient that they distribute weight perfectly from heel strike to toe off. The rest of us depend on our running shoes to neutralize the flaws in our sadly imperfect feet. Understanding the structure of your foot will help you understand the type of running shoe you need.
You can easily determine your foot's general type with two simple tests. This first diagram should help you decipher the wear patterns of your running shoes:
To double-check, give yourself the tried-and-true "bathroom test." When you get out of the shower with wet bare feet, take a look at your footprint:
The over-pronator. You over-pronate when your feet roll in when you run, and you supinate when your feet roll out. It's natural to pronate somewhat when you run, but many runners have feet that simply overdo it. Take a look at the bottom of your running shoes. If you are an over-pronator your shoes will be more worn on the inside edge of your shoes than the outside edge. In severe cases, the shoe will actually slope dramatically inward. You may have trouble with shin splints or runner's knee. Over-pronation is a symptom of highly flexible feet that need a little help from their running shoes to maintain stability. For added motion control in your shoe, you will generally have to sacrifice a certain amount of cushioning. Rest assured, it's worth it.
Specifically, you should look for a shoe with a firm midsole, especially on the inner side of the shoe. The shoe should have a straight board last and a rigid, durable heel counter. A good arch support might be helpful.