US Army Rangers, Green Berets and Delta Force all report to the Special Operations Command. They each carry out similar duties, but the level of training required for each varies as does the selection process.
US Army Rangers are Ranger qualified and wear a "Ranger" tab on their uniform. They are basically advanced infantry and are utilized in large size units of platoons and organized into Ranger Battalions. The selection process is difficult and requires passing a three-phase course based out of Ft. Benning, GA.
US Special Forces, also known as Green Berets, are Special Forces qualified and wear a "Special Forces" tab on their uniform. They generally fight in smaller units of squads or teams. The selection process is probably more rigorous than Ranger school and is a two tiered system. First you pass the entrance test before taking the qualification course.
US Delta Force typically avoid wearing uniforms. They attack in small, 12-man teams and focus on counterterrorism tasks. They pick their members, about 50 per year on a good year, from the Rangers and Green Berets by invitation and selection. They are seen as the cream of the crop of Army special warfare units.
They are elite for three reasons. First, their training sets them apart from traditional military units. Each member may have a specialty, but all will have functional knowledge of every piece of equipment, be jump qualified, often attend special schools such as SERE and combat diver. Secondly, the physical standards are so high these men are physically capable of outperforming the average soldier. They can carry more and go through more demanding scenarios. Finally, they receive the best equipment, some of which is still in the experimental stages, to achieve their mission. Their weapons will be more compact, lighter and have more attachments than the standard M-16 or M-4. They have highly capable vehicles, radios and other ordinance as well, which allow them to take on extremely dangerous scenarios.