All are IV administration methods. A piggyback is when the medication you are wanting to give has been mixed into a small plastic container of IV fluid, usually 50 or 100mL of 5% Dextrose in Water or 0.9% saline (normal saline). A tubing is then attached to this small bag and it is hung higher than a mainline bag of IV fluids (for gravity purposes) and allowed to infuse into the patient over a determined amount of time, often 30minutes to an hour. Because it is hanging above the main IV bag, it overrides the infusion of the main IV bag. The nice thing about it is, when this smaller bag is empty, the main IV bag will then automatically start to drip again on it's own. So, one of their advantages is that they are kind of a time-saver method of IV medication administration.
A Buretrol is a special type of IV tubing that contains a tubular section called a buretrol which can hold about 150cc of IV fluid. This buretrol tubing is connected to a large bag of IV fluid (500 or 1000mL of 5% Dextrose in Water or 0.9% saline). Then the buretrol tubing can be connected to a mainline IV tubing similar to IV piggyback tubing, or they can also be connected intermittently directly to the patient's saline lock. When you are ready to administer an IV medication, you open a clamp on the tubing that allows the buretrol to fill with the amount of IV fluid you wish. You then inject the IV medication into the fluid in the buretrol at an injection port at the top of the buretrol. Similar to piggybacks, you open the IV roller clamp below the buretrol and allow the admixture that you have just mixed to infuse into the patient's main IV line or saline lock. The drip chambers of buretrols often deliver microdrops. The other use for them is as a protection against accidently allowing too much IV solution to infuse into a patient over a short period of time, often babies. An amount of IV fluid can be allowed to enter the buretrol and then infuse from the buretrol. The inconvenience of this is that you have to periodically check to make sure your buretrol hasn't gone dry. The advantage is that when used as a gravity drip (no IV pump) you have better control of the amount and rate of IV fluid the patient is going to get. In today's world of IV pumps I doubt that you will see many buretrol tubings in use. Years ago, buretrols were very commonly used with small children and babies.
An IV push is when you load the IV medication you are giving into a syringe, access the IV line yourself and literally push the medication into the patient. It is often done as close to the IV site as possible and there are guidelines for most IV push medications as to how fast over a period of time you can inject them. These days mini-infusers (piston driven infusion pumps) have come into common use so nurses don't have to spend time at the bedside administering IV push medication while watching her watch (also another time-saver for the nurses).
An IV bolus is pretty much the same as an IV push. The medication is given in one shot, no mixing with any other IV fluid. So, when you are giving a bolus, the main IV tubing is stopped and not re-started until the bolus is completely injected. During code blues most of the IV medications are given as boluses.