These are chancery court cases.
_Ads_ is an abbreviation for _ad sectam_ , Latin for "at the suit of". The alternative is _vs_ for _versus_.
The difference is that for papers associated with the defendant entering the plea, the names are written [defendant] _ads_ [plaintiff]. For papers associated with the plaintiff filing charges, the names are written [plaintiff] _vs_ [defendant].
For example, the first file on the snippet given, Shannon _ads._ Newman, Joseph Shannon is the defendant at the suit of plaintiff Alex Newman.