You will most likely want a motion control shoe with extra support. Here is an article from runnersworld regarding the basics of shoe selection and how to self diagnose neutral, over pronator or underpronator (supinator - much rarer)
As a heavier runner you will want to look for shoes that talk support and stability. Online reviews of shoes often talk about the weight of the shoe. Beginning runners do not need to worry about this, that is for racing, and in general lighter shoes will not have enough support for heavier runners.
There are many brands that make good running shoes, but different feet just fit different shoes. For example, before I had children Nike's fit my feet very well, now they do not fit at all and Asics & Brooks fit better. If you buy online look for a place with a good exchange policy and do not be afraid to return them if they do not feel comfortable or you get blistering or other problems.
Sizing is the biggest issue. MOST people will buy their shoes too small. Running shoes are quite often about 1/2 size larger than dress shoes.