Volts is a measure of pressure and it's volts that drives a current, measured in amps or milliamps, through your body when you make contact with electrical conductors. Current equals voltage divided by resistance, but the total resistance is influenced by several factors. These include the resistance of the bulk of your body through which current flows, whether your skin is moist or dry, the texture of your skin (i.e smooth or calloused) and the shape of the conductors in contact with skin. Pointed electrodes can penetrate into skin and make better contact and also greater surface area of contact will reduce resistance. If current flows through your chest, it can cause fibrillation of the heart or cardiac arrest. Fibrillation is when the muscle fibres of the heart are all contracting randomly rather than in sync, so they don't pump blood properly.
Is it volts or amps that kills? Volts drives the current, but it's current that kills. A current as low as 30mA can cause ventricular fibrillation, so RCDs (residual current devices) that detect earth leakage faults are designed to trip and shut off power when a current of 30 mA flows (e.g. if you touch a power cable).