Last Night on Cops in Fort Worth Texas there was a foot chase with a bad guy and one of the back up officers who responded was a Fort Worth "Deputy Marshall". Her shoulder patch even said "Marshall". Can someone explain this? How does it compare to patrolman?

City Marshall.  We have them too.  They work warrants.

A friend of mine is a marshall for a small city.  From what I understand, they have full arrest powers.  

Usually they are warrant officers for Municipal Courts.

Legally, a Marshal (one "L") has the same status as a Police Officer in Texas when commissioned by an incorporated city, town, or village. Historically, they are in separate departments in the larger cities and generally serve Municipal arrest warrants (typically traffic offenses) and sometimes handle bailiff duties. I recall at least one smaller town who designated all of their municipal Peace Officers as "Marshals."

Check out Chapter 2, Article 2.12, section 3 of the Texas CCP for more information.