Volcanic bombs or lava bombs are globs of magma thrown out of a volcano during eruption. Volcanic bombs are distinguished from other types of tephra by their large size and the fact that they are molten when they are expelled.
Generally tephra is classified as a volcanic bomb if it is greater than 64mm at its greatest diameter. This is the largest of the volcanic ejaculate, smaller sizes being referred to as lapilli (small stones and pebbles) or ash (dust and small particles less than 2 mm in diameter). There are some sources that give the required size for bombs as only 32 mm, but these sources tend to be older; it is now usual to call debris from 32 to 64 mm in diameter 'coarse lapilli' or simply 'lapilli'.
Volcanic bombs are also required to be somewhat fluid (partially or entirely plastic) when they leave the volcanic vent. If a hardened rock is thrown into the air by the blast it is called a block, not a bomb. Because bombs are plastic when they are launched they may take on many shapes, often somewhat aerodynamic in nature. Bombs are often classified by their shape.
Ribbon bombs are formed by fluid lava that has been ejected in long streamers.
Spherical bombs form from particularly fluid magma, in which surface tension can play a large part in shaping the projectile.
Spindle bombs (AKA fusiform, almond, and rotational bombs) are also formed from particularly fluid lava, but in this case the rotational force imparted on the bomb gives it an elongated shape.
Cow pie bombs are spherical or spindle bombs that did not solidify before impact, and flattened or splattered when they landed, forming flattened disks.
Bread-crust bombs are formed when the outer shell of a lava bomb cools and starts to solidify during flight, but the inner core continues to expand; the expanding interior will crack the crust, looking not unlike a well-baked loaf of bread.
Cauliflower bombs occur when a bomb falls into water, causing the bomb to expand suddenly and unevenly, making a formation much like pillow lava.
Cored bombs are bombs that have a crust of lava surrounding a core of native rock or older volcanic rock that was sitting on the surface of the volcano when it erupted (this non-molten rock is known as accidental lapilli).
Volcanic bombs may actually explode, both when landing (essentially 'bursting' on impact and splattering lava everywhere), or minutes after landing, as the gasses in the hot inner core expand against the cooled outer shell (although this is rare). Most damage caused by bombs comes from either the direct impact of the bomb, or from igniting wildfires. Giant projectiles weighing tonnes may travel up to a third of a kilometer before impact, and smaller ones weighing only a few kilograms may travel up to 6 kilometers. You are unlikely to be hit by a bomb unless you are very near the volcano, however. The amount of tephra hitting a given patch of land falls off as a square of the distance from the eruption.
Volcanic bombs are most often associated with strombolian, vulcanian, and Hawaiian eruption types. Volcanic bombs may be of particular interest to rock hounds because you will occasionally find one whose interior contains crystals, most notably olivine.
Pictures can be found here and here. Bombs containing olivine can be seen here and here.