Army: The main ground-force of the United States. The Army's main function is to protect and defend the United States (and its interests) by way of ground troops, armor (tanks), artillery, attack helicopters, tactical nuclear weapons, etc.
Air Force: (The baby branch with the most money) The primary mission of the Air Force is to defend the United States (and its interests) through exploitation of air and space. To accomplish this mission, the Air Force operates fighter aircraft, tanker aircraft, light and heavy bomber aircraft, transport aircraft, and helicopters (which are used mainly for rescue of downed-aircrew, and special operations missions). The Air Force is also responsible for all military satellites, and controls all of our Nation's strategic nuclear ballistic missiles.
Navy: The Navy's primary mission is to maintain the freedom of the seas. The Navy makes it possible for the United States to use the seas where and when our national interests require it. In addition, in times of conflict, the Navy helps to supplement Air Force air power. Navy aircraft carriers can often deploy to areas where fixed runways are impossible. An aircraft carrier usually carries about 80 aircraft. Most of these are fighters or fighter-bombers. Additionally, Navy ships can attack land targets from miles away (with very heavy guns), and cruise missiles. Navy submarines (fast attack and ballistic missile subs) allow stealth attacks on our enemies from right off their shores. The Navy is also primarily responsible for transporting Marines to areas of conflict.
Marine Corps: Marines specialize in amphibious operations. In other words, their primary specialty is to assault, capture, and control "beach heads," which then provide a route to attack the enemy from almost any direction. While amphibious operations are their primary specialty, in recent years, the Marines have expanded other ground-combat operations, as well. The Marines are generally a "lighter" force when compared to the Army, so they can generally be deployed fast (although the Army has been making great strides in "rapid deployment" in the past few years). For combat operations, the Marines like to be self-sufficient, as much as possible, so they also have their own air power, consisting primarily of fighter and fighter/bomber aircraft and attack helicopters. Even so, the Marines use the Navy for much of their logistical and administrative support. For example, there are no doctors, nurses, or enlisted medics in the Marine Corps.