Becoming a Registered Nurse simply means sitting for, and passing the National Council Licensure examination (NCLEX), which is a type of examination that the Board of Nursing in every state uses to determine whether a candidate is qualified to practice as a Registered Nurse or otherwise.

You can only take this test after you have completed a Degree program in Nursing, so technically, it is impossible to become a Registered Nurse without a Nursing Degree.
However, there are other courses and programs that you can take that qualify you to take the NCLEX exams even if you don’t have a Bachelor’s degree.

Associate Degree in Nursing: Associate Nursing Degree take less time to complete than a Bachelor’s degree. In 2 years, you are done with your associate degree program whereas a bachelor’s degree program would take no less than 4 years to complete.
Associate Nursing degree programs are widely available in career centers and community colleges all over the country. In fact, according to the National Organization for Associate Degree in Nursing, there are more than 1,000 of such programs in the country so it’s easy to enroll in one and obtain a degree after which you would be qualified to sit for the NCLEX examination that would qualify you to become a Registered Nurse.
Hospital-based Diplomas: Another way to become a Registered Nurse without a Bachelor’s degree is to complete a 3-year hospital-based nursing program. This type of Diploma program dates back to the early 19th century and is very popular amongst nurses.
Hospitals that offer these programs are affiliated with universities and licensed to offer diploma-based programs. In some cases, you will be able to work and learn at the same time.
On completion of your 3-year diploma program, you’ll be allowed to write the NCLEX examination and qualify as a Registered Nurse.

Apart from these 2 methods, the only way to qualify as a Registered Nurse is to either obtain a Bachelor’s Degree in Nursing, or a Master’s Degree in Nursing.