There is one major difference between all Asian pear varieties and European pear varieties: how they ripen. European pears must be picked and sent to market while they are not yet ripe. Leaving them on a counter at room temperature allows them to ripen, much like bananas or apples. Asian pears, however, must be fully ripe before they are picked. They will not ripen any further once they have been removed from the tree.
Whether buying Asian pears in a supermarket or harvesting them from your own trees, there are a few indicators that you should pay attention to in order to tell if they are ripe. Asian pears also have other names, including Japanese pear, African pear, Nashi pear, Nashi apple, Korean pear, Taiwan pear, bae li, bapple, sand pear, pear apple and papple. They take the apple parts of these names from their round shape rather the usual pear shape. Know your pear variety.
In order to produce good local pears, producers depend on ideal spring and early summer weather conditions, and no late frosts. Now, here's the surprise: pears are picked unripe and left to ripe in a cool, dry, dark place like a basement or garage. If you wait for them to ripen on the tree, you probably won't harvest many - they'll rot and be attacked by bugs and birds. See this page for descriptions of pear varieties.
You are going to hear lots of conflicting stories about Asian Pears over the next few decades. They are tart, they are sweet; they are very good to cook with, they are not good to cook with; the trees are self-pollinating, they are not self-pollinating. And, all these stories are true. There are twenty very different ones listed on this site alone.