Ava,
An MSW is what you receive after graduating from social work school (at a Master's level) . Depending on the state you live in there are different pathways AFTER receiving your MSW...one of those is to become an LCSW... Usually after graduation and receiving an MSW, you will work with a LIMITED license (LLCSW) for 2 years and then take an exam which makes you a fully licensed clinical social worker. You would be able to diagnosis and treat (therapy, etc.) and have a private practice, but not prescribe medication. You could graduate from social work school and decide to never become licensed...and you would STILL have an MSW. Just like you could get a bachelors in engineering and never become an engineer. You will always have the degree...However, you MUST have an MSW to eventually obtain an LCSW. At least in my state.
On a side note, if you are thinking of doing therapy, I will say there are a lot of things to consider when choosing social work as a path. There are other options which I believe prepare people much greater for clinical therapy than social work - such as becoming a psychologist or a licensed counselor. You will gain more "therapy" skills in those programs, however, you will not address systems views as much or examine social justice related topics, oppression, power dynamics...But you aren't going to perfect your DBT and CBT skills (therapy modalities) in social work school to the extent you would getting a masters in counseling or a doctorate in clinical psychology. If you are wanting to do psychological research...if that is what you mean by psychological testing...then social work is not the field for you and I would probably pursue a doctorate in non-clinical (research) psychology or maybe psychiatry. Most people who are paid to do research in the U.S. have PhDs and many work for Universities. You could become a researcher doing psychological testing as a social worker, but the path would be incredibly uphill and probably require a lot more luck.
Hope this is helpful!