So think of an ADC as part servant, staff officer, part adjutant. Basically they were assigned to the Flag Officer to make sure they could do their job, be it by managing correspondence, retrieving supplies/securing housing, running messages, or serving as the General's eyes on distant parts of the field.

Typically they would be young LT's or Captains who migthve come from influence, or caught the eye of the General. This was before the modern staff system had been adopted so these men very much were responsible for the ability of the unit to function and commanders to communicate.

As far as uniforms go what I have read implies that they would have been attired just like any other field grade officer, which in and of itself was quite broad. Simply put they rigors of active service made it so that they would have had nothing much more then the basic officers uniform. And that was also usually at the discretion of their commander, Grant was known to sometimes wear a simple private's coat with ranks on it as it was simpler then the officer's jacket.

Hope this helps. Let me know if you have anything more specific

To follow up, what exactly is an aide-de-camp? My dad has a letter on his office wall from the governor making him a colonel aide-de-camp, but I never have gotten a clear explanation as to what that means.