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.
OA. I would suggest that depending your ocupation, if pain is elicit with grabbing such as stiffness one of the most common places of affection in osteoarthritis is the base of the thumb. Pain and stiffness are some of the symptoms and a simple x rays of the hand can make the diagnosis.
Could be many things. This could be the first sign of arthritis or could be an early trigger finger.
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.