A high TSH and low T4 is a diagnosis of primary hypothyroidism. There are various causes of hypothyroidism but the most common cause in developed countries is an autoimmune disease called Hashimoto's thyroiditis.
Testing for Hashimoto's thyroiditis antibodies: thyroid peroxidase antibodies (TPOAb) and thyroglobulin antibodies (TgAb). If you have Hashimoto's thyroiditis you may experience both hypothyroid and hyperthyroid symptoms (Hashitoxicosis). I posted more info about the two types of Hashitoxicosis on another answer...
http://www.medhelp.org/posts/Thyroid-Disorders/Hashitoxicosis/show/2719050