A small set of tools handles 95% of fragging. A bone-saw or Dremel with a diamond cutting wheel for stony coral, a pair of wire cutters for branching coral, sharp scissors or a scalpel for soft coral, reef-safe cyanoacrylate gel (Bob Smith Industries or similar), and an assortment of frag plugs or rubble pieces for mounting. Add a coral dip (CoralRx, Bayer, Revive) to the kit for treating frags before they go in the display.
For Acropora and Montipora, cut just below a branching point with the diamond wheel. Aim for a frag about 1-2 inches long. Mount immediately with cyanoacrylate gel on a frag plug, base down, and place in low-to-moderate flow with moderate light. Healing time is 2-3 weeks - you will see the cut site encrust over and tissue grow across the wound when the frag has accepted the mount. Acropora is the most temperamental SPS; mount and walk away, do not move it.
Euphyllia (Torch, Hammer, Frogspawn) frag along the natural branch divisions. Use the bone-saw to cut at the base of a head, leaving most of the skeleton intact. Mount the frag immediately and place in moderate flow with moderate light. Acanthastrea and Lobophyllia frag along the visible polyp boundaries with a Dremel - aim to keep at least one healthy mouth per frag. Goniopora is the hardest LPS to frag; if you can avoid it, do.
Sinularia, Sarcophyton, Cladiella, and Lobophytum frag with sharp scissors or a scalpel - cut a small lobe or finger off the parent colony. Soft coral frags do not bond to plugs the way SPS does, so the standard mounting technique is to use a small cyanoacrylate dot to stick the cut surface to a piece of rubble, then secure the rubble with a rubber band or fishing line until the frag attaches naturally (3-7 days). Zoanthids frag by cutting the substrate the polyps are attached to with a Dremel - never cut directly through a polyp.
The first 48 hours after fragging are the highest-risk period. Cut surfaces are exposed to bacteria; the coral is energetically depleted. Drop the frag in low-flow, moderate-light location and resist the urge to move it. Watch for tissue recession around the cut - normal recession is less than 1 mm; deeper than 2 mm means the frag is failing and may need to be cut back further into healthy tissue. Polyp extension within 5-7 days is the strongest sign of survival.