It depends on how you define each of those terms and what your
     particular values are. Rather than try to make a broad
     generalization, I'll describe the products and practices that are
     usually called into question regarding these topics. You are then
     free to apply these facts to your own system of beliefs and make an
     informed judgement. Also, I have ignored the fact that beer is an
     alcoholic beverage produced by the metabolism of yeast. This should
     be taken for granted. Read labels carefully and call the brewer if
     you need specific information about ingredients or processing since
     labeling laws allow the brewer to omit a great deal.

     Finings
          Finings are substances sometimes added to beer during
          fermentation to help settle out particles and yeast, leaving the
          beer clear. It is important to note that finings are not present
          in the finished beer in any significant quantity. Their purpose
          is to settle out of the beer, not stay in suspension. OTOH, if a
          careful chemical analysis were to be performed, there would
          probably be a few molecules of a fining agent still to be found.
          Also, many brewers do not use finings at all, but filter their
          beer to clarify it. That said, these are the common fining
          agents:

          Isinglass
               Made from the dried swim bladders of sturgeons. Used a
               great deal in British brewing.
          Irish Moss
               Also known as carragheen, a type of dried seaweed.
          Gelatin
               The same stuff used to make Jello (tm). Made from animal
               (mostly cow) hooves, skin and connective tissues.
          Polyclar
               A brand name for PVP (polyvinylpyrdlidone), a man-made,
               plastic substance.
          Sparkalloid
               More commonly known as diatemaceous earth.

          FYI, beer brewed according to the Reinheitsgebot (see related
          Q&A) is not prohibited from using finings since it was generally
          assumed that finings were not present in the finished product.

     Adjuncts
          These are products used to alter the flavor, color, or body of
          beer. They are used in addition to the "Basic 4": malted barley,
          hops, yeast, and water. They do not settle out and can be
          present in beer in significant quantities.

          Corn
               Used a great deal by the mega-brewers as a cheap way to
               make huge quantities of beer since corn is cheaper than
               malted barley.
          Rice
               Same as corn.
          Wheat
               Used in some beer styles to produce a lighter-bodied beer
               with a tangy flavor.
          Honey
               Used as another fermentable sugar in addition to malted
               barley to impart different flavors.
          Lactose
               Also known as milk sugar because of its dairy origin. Used
               to increase sweetness and body of certain beer styles such
               as cream stouts.
          Molasses
               Another form of sugar used to flavor some dark ales.

     Heading agents
          Various products added to a beer to increase its ability to form
          and hold a head. Used most often in beers made with large
          quantities of corn and/or rice. Pepsin is a common heading agent
          and is often derived from pork. Beers using only malted barley
          or wheat don't need heading agents.

     Organic ingredients
          To be truly organic, a beer would have to be made from barley
          and hops cultivated using accepted organic practices. Most
          brewers do not make this claim, but a few are appearing. Those
          that do clearly label their products as organic. It is also my
          understanding that organic does not mean no animal products.

     Other ingredients
          Many other ingredients are used in brewing beer to give it
          unusual character or marketing appeal. As such, these items are
          often clearly indicated on the label. Some of the more common
          examples are:

          Oatmeal, Pumpkin, Potatoes, and all sorts of fruit
          Also spices such as: Ginger, Licorice, Coriander, Cinnamon, and
          Spruce