The death penalty is a very controversial and emotional topic, so I will try to keep it mostly focused on the facts of the various methods that are used today.
There are 5 different legal modern methods of execution used in the United States; in order of most commonly used they are: Lethal Injection, Electrocution, Lethal Gas, Hanging, and Firing Squad (both coming in at .2% of executions since 1976).
1. Lethal Injection
In 1982, Texas became the first state to administer lethal injection, operating on Charlie Brooks, who was convicted of the kidnapping and murder of a car sales man named David Greggory. Now both every state and the federal government use lethal injection as their primary method of execution. Almost all states use a three step process, in which they first administer a drug, commonly sodium thiopental or sodium pentothal, that renders the convict unconscious. Then they administer pancuronium bromide, which is a muscle relaxant that paralyzes the diaphragm and lungs. And then, finally, they induce cardiac arrest with potassium chloride. It is believed to be very efficient, but there have been flaws exposed that are slightly terrifying. In 2005, the "Lancet Journal" released a report about how many of the cases (21 / 43%) in southern/western states showed that a convict had inadequate anesthesia that wore off prior to the third dosage Therefore, this left feeling excruciating pain prior to death. Of course this was back in 2005, in a select few states that did not have strict requirements for the doctors who administered the dosages, so overall lethal injection is still considered very reliable and efficient.
2. Electrocution
When a person is incapable of receiving lethal injection (when they have collapsing veins or a condition or choose to face another type) that is when the other methods are used. In 1890, the first execution by electrocution in history was carried out against William Kemmler, who had been convicted of murdering his lover, Matilda Ziegler, with an axe. And through out the majority of the twentieth century it was the main form of execution, executing many notorious convicts, such as John Arthur Spenkelink, Bruno Hauptmann, or Ted Bundy. Today 9 states (Alabama, Arkansas, Florida, Nebraska, Ohio, Oklahoma, South Carolina, Tennessee and Virginia) provide electrocution as a viable alternative to lethal injection.
3. The Gas Chamber
The next most used method is the gas chamber, which has been used 11 times since 1976. The convict is sat down and restrained in an air tight chamber, then three executors turn their keys simultaneously, which releases cyanide pellets into a sulfric acid solution, causing a lethal gas to flood into the room. It was first used in Nevada in 1924, and is currently an option in 4 states (California, Wyoming, Arizona, and Missouri).
4. Hanging
Prior to the 20th Century, hanging was the most common form of execution in the United States. Now in modern times, there have been only 3 executions by hanging since 1977 (the most recent on Billy Bailey in 1996). The procedure is slightly self explanatory, but full of complex and necessary calculations to account for height and weight of the convict to make sure that the regulation, of 1200 pounds of force, is applied to the neck.
5) Firing Squad
Unlike in China, where a single executioner fires from a short range into the back of the head of the condemned, the United States has a little bit more complicated process. There are three to six shooters who stand behind a wall, with slots in it, 20 feet away from the restrained convict. Once the command is given they all fire at the chest. An interesting fact about the use of firing squad, is that one of the shooters is firing a blank instead of an actual bullet, but none of the executioners know which one of them has the blank. The purpose is to ease their consciences, because they can have hope that they had the blank and consequently didn't kill anybody. Only 2 states (Oklahoma and Utah) currently allow shooting as a method of execution, and since 1976, only 3 (0.2%) of the 1,389 convicted murderers executed were by firing squad.