Throughout the the United States and elsewhere in the world that American low-and slow barbecue has caught on, it is customary to smoke with local hardwood, for obvious reasons. In Montreal they smoke with maple, in Georgia peach, Tennessee they like hickory, oak is popular in upper midwest, and in South Dakota they use corn cobs!
Here in Arizona and other parts of the desert southwest, we generally use mesquite as it is inexpensive and plentiful. Mesquite wood, however can be an overpowering flavor, and it is easy to oversmoke with it, making the food taste like little but campfire.
When smoking with mesquite wood, it is best to use wood with some 'season' on it. That is, some wood that has been aged for a few months, up to a year, and is not' green', or freshly cut. The fresher the wood, the stronger the smoke flavor, so the less wood you need.
Often, I find myself smoking with simply mesquite charcoal, which is little more than partly burnt pieces of mesquite wood. Being partly charred already tends to remove some of the overpowering mesquite flavor from the wood that many find objectionable, and its much harder to oversmoke with it.
Should I crave a different smoke flavor, I will use a neutral charcoal briquette as opposed to hardwood charcoal, and than add the apple, cherry, oak, etc....after the coals are glowing.
Mesquite is a truly authentic flavor of the desert Southwest, just don't overdo it!