The connecting rod connects the pistons to the crankshaft. It converts the linear motion of the pistons to the rotary motion of the crankshaft. On every stroke, the connecting rod is stretched and compressed. This pressure, plus other factors, can cause the connecting rod to break. The broken rod can go through the engine block completely, ruining the engine--a condition known as "throwing a rod."
Fatigue is the main cause of broken connecting rods--especially in older engines. The constant compression during the power stroke and stretching during the exhaust stroke, over thousands of times a minute, eventually wears the metal out and it becomes brittle and finally breaks. If the oil is low or dirty it can speed up this process. Running the engine hot can also speed up the process. Sometimes a fairly new engine can have fatigued connecting rods if it is a rebuilt engine and the mechanic used cheap parts or the wrong parts for the engine.
The pin that connects the connecting rod to the piston (called the piston pin, wrist pin or gudgeon pin) gets a lot of wear. If this pin snaps the connecting rod is no longer connected to the engine. For some engines this results in catastrophic engine failure--the connecting rod goes through the engine block or the crankshaft is bent--but for some engines it just causes a dramatic loss of power. If the engine is stopped immediately after the pin breaks it might be possible to save the engine.
Over revving is the main cause of connecting rod failures in new and high performance engines. If the tachometer hits the red--even briefly--the connection rods are in danger of breaking. This is because the forces acting on a connecting rod increase dramatically at high revolutions. It does not matter if the tachometer is going into the red because the car is travelling at a high speed, is going too fast in a low gear or is simply going too fast because the accelerator is pressed too far while the car is in neutral--the stress is simply too high at extremely high RPMs.
Hydrolock is a deformation of the connecting rod caused when water gets into the piston chamber. This usually happens after the car has been driven through deep water such as a flooded street. If only a little water gets into the cylinder the car makes a knocking or tapping sound and it can be repaired (have the water taken out and the gaskets replaced), but if enough water gets in the cylinder that it takes up all the space available at spark time, the connecting rod will bend or snap. Hydrolock is much more common in boats than in cars because boats are always operated around water.