What do the terms 'holding out' and 'common carriage' mean?
Oddly enough, "common carriage" is not defined in the FAR's, but that is because it is a "common law term" and not specific to aviation.

AC120-12A - Private Carriage Versus Common Carriage of Persons or Property contains guidance that can be used to help determine whether or not an operation falls under common carriage.  Take a look at it for specific examples, but the "plain language" version is:

Common Carriage: A carrier becomes a common carrier when it "holds itself out" to the public, or to a segment of the public.

Holding Out:  A carrier is holding out when they represent themselves as willing to furnish transportation within the limits of its facilities to any person who wants it.

There are lots of ways to "hold out", but the basic idea is if someone in the general public comes to you and asks you to fly them and you agree, you are probably holding out.  Any form of advertising demonstrates a clear sign that you are holding out.  There are also other activities that qualify, so look over the AC if you have further questions, and if that doesn't answer it your best bet is to contact an aviation lawyer.  You could also contact your local FSDO with questions.



As far as transporting a third part for hire, both common and private carriage require an air carrier certificate issued under 125 or 135 depending on the size of the airplane1. This means that your trip for a friend would be in violation because you don't have the required air carrier operating certificate.  (Thank you @p1l0t for clarifying this for me!)

119.23 says:


  Operators engaged in passenger-carrying operations, cargo operations,
  or both with airplanes when common carriage is not involved.
  
  (a) Each person who conducts operations when common carriage is not
  involved with airplanes having a passenger-seat configuration of 20
  seats or more, excluding each crewmember seat, or a payload capacity
  of 6,000 pounds or more, shall, unless deviation authority is issued--
  
  (1) Comply with the certification and operations specifications
  requirements of part 125 of this chapter;
  
  (2) Conduct its operations with those airplanes in accordance with the
  requirements of part 125 of this chapter; and
  
  (3) Be issued operations specifications in accordance with those
  requirements.
  
  (b) Each person who conducts noncommon carriage (except as provided in
  Sec. 91.501 (b) of this chapter) or private carriage operations for
  compensation or hire with airplanes having a passenger-seat
  configuration of less than 20 seats, excluding each crewmember seat,
  and a payload capacity of less than 6,000 pounds shall--
  
  (1) Comply with the certification and operations specifications
  requirements in subpart C of this part;
  
  (2) Conduct those operations in accordance with the requirements of
  part 135 of this chapter, except for those requirements applicable
  only to commuter operations; and
  
  (3) Be issued operations specifications in accordance with those
  requirements.


The FAA takes a very dim view of people who start competing with actual air carriers and are unlicensed to do so.  It's a quick way to have your license suspended and end up with some hefty fines and other legal troubles.



1 There are of course the regulations that allow operations "in furtherance of a business" which is why I specified a third party.