I have been running for many years and have tried just as many brands.
I you have narrower feet and the best brands for you are Adidas. If you have normal width feet Asics, Brooks and Nike will be right for you.
New Balance are for those with very wide feet Brooks also put out a shoes for those with wider feet. I have only include shoes that some of my running friends uses.
I would recommend that you should go and see a podiatrist before you run. This will assist in finding out if you have any lower leg problems. Find out if you have high arches or normal or even flat feet. This may change the type of running shoe you may buy.
Shoes with motion control are also good. Each company call their motion control something else, so check to find out what they call it. Also ask your running shoes specialist what they call it or check out each company website for details.