For a disability decision, if for example a veteran does not agree with the decision of the department regarding a disability benefit, the veteran could enlist the assistance of the Bureau of Pensions Advocates, which provides free legal assistance to veterans to appeal disability benefit decisions. There are offices across the country. They will help the veterans to appeal. The process then takes place before the Veterans Review and Appeal Board panel, but there are levels of appeals. There's a departmental review, a review hearing, and there's also an appeal hearing.
If you want more detail regarding the inner workings, I would refer the question to either a representative from BPA or from the Veterans Review and Appeal Board. They could provide more details regarding the process.