A Research Chemist typically uses their chemical and industry knowledge to try to create new things (this is usually called Research and Development) and then try to determine the stability and effectiveness of what they created. Examples include: drug research, fuel research, and private products (ie: cleaning solutions, vitamins, food, sodas, ect)
A testing chemist, often called an Analytical Chemist, uses their chemical knowledge to identify and quantify materials using wet chemistry (concept of using chemical reactions and chemicals to accomplish this) and/or instrumentation (there are lots of instruments to help depending on the industry/purpose the most common are FTIR, LC-MS, ICP-MS, GC-MS, and Autotitrators). This is probably the most diverse field and includes lots of careers such as Forensics, medical testing , quality testing, mining, water chemists, food chemists, public health ect.
I am a quality chemist which means that at my company I determine the safety, content, and quality of our products using chemical principles and instrumentation.
The last type of chemist uses their chemical knowledge to make instruments and testing methods to help the first two type of chemists. Most of these people work at governing bodies or companies that make analytical instruments IE: ACS, USP, Waters, Agilent. These are the most competitive type of jobs are requires the most experience and knowledge.