I've been having pain on my hand between thumb and index finger on the fatty part. I could feel the pain under my thumb inside my hand. Hurts to hold?
Hand pain. Hands take a lot of abuse daily and there are several muscles and ligaments that traverse this area and are at risk for injury, strain and tendonitis. The thumb has a particularly complex articulation (the way the joint and associated muscles assemble and function) and is a common site for problems. If painful, prolonged or debilitating, have it evaluated by your physician.
Gall bladder issues. People who have gallbladder issues will have pain near thumb fatty pad - treat locally think globally.
Arthritis. Arthritis at the base of the thumb gives the pains you describe. An x-ray can easily help identify it. A small thumb brace can be very helpful. Surgery is also an excellent fix if the case is bad.