It is important to reiterate that the antibodies themselves do not destroy anything. When an antibody bonds to an antigen, multiple things can happen, including complement fixation and activation, neutralization, agglutination, precipitation, and clonal selection. The last one will be described in the Antibody Production and Immunological Memory section. All of these methods can be used on the same antigen at once.
This result is generally experienced when an antigen protruding from a cell/virus is bonded to by an antibody, usually those of the IgM or IgG type for the Classical Activation Pathway, which involves infected human cells, and IgA or IgE for the alternate pathway, which involves bacteria and viruses (http://www.cehs.siu.edu/fix/medmicro/cment.htm). When cells produce proteins, they stick small pieces of it out of their membranes. If the proteins are antigenic, antibodies bond with them. When an antibody bonds, the crystallizable fragment changes shape to reveal a compliment binding sites which, the C1 enzyme can bond with (http://www.cehs.siu.edu/fix/medmicro/cment.htm). When it does, the complement cascade begins. This ultimately results in the formation of the complement membrane attack complex, which causes cell lysis (it pokes holes in the cell) (http://www.cehs.siu.edu/fix/medmicro/cment.htm). Additionally, this process draws attention to the cell, aiding in phagocytosis (getting swallowed by a phagocyte) (http://www.cehs.siu.edu/fix/medmicro/cment.htm). For a more complete description of the complement cascade, check out this page from the Southern University of Indiana Carbondale.
Neutralization is not something that happens after bonding. It is a direct result of the bonding itself. In neutralization, the antibodies bond to dangerous areas of antigens, effectively sealing them off and rendering them harmless. In cases where the majority of an antigen poses a threat, it may become almost entirely surrounded by antibodies. Once neutralized, antigens are easily identified by phagocytes and destroyed.
In agglutination, a single antibody bonds with multiple antigens, 'gluing; them together. Also, multiple antibodies are often involved, creating large chains/clumps (Marieb). This facilitates phagocytosis in multiple ways: the lymphocytes have an easier time finding the larger target, the lymphocytes can engulf them all together, and the lymphocytes do not have to chase after mobile antigens such as some bacteria, because they are much slower/immobile when 'glued' together.
Precipitation is similar to agglutination. Sometimes, antigens are soluble. In precipitation, antibodies form long chains of antibodies and antigens. The goal of it is to cause the soluble substance to precipitate out of the solution (Marieb). Once they are out, phagocytes have an easier time engulfing them, because they are no longer dissolved, and because of the reasons listed for agglutination.