Answer:
Modern submarines (true submarines, not Deep Submergence Vehicles, which rely on a support tender and run on batteries charged by the tender) are generally of two types, Nuclear and Diesel-Electric.
Nuclear Power is the preferred method of propulsion for most modern submarines, since their speed, range, and diving capability outweigh the advantages of a DE boat, which is quieter when running on battery with its electric motor. However, all DE's must eventually recharge their batteries with their diesel engines (even modern boats who can go weeks without a battery charge), and that key tactical limitation is what gives Nuclear boats the overall advantage.
When the debate concerning the conversion to Nuclear Power or staying with DE technology (the cost for NP is far greater, but so is the capability), a test was devised where one nuclear submarine was pitted against several DE submarines. The rest is history - each DE submarine was detected and "sunk" via simulated attack when it had to come to periscope depth and snorkel to recharge its batteries. Since that time, the U.S Navy now exclusively builds nuclear submarines, the last DE boat being retired almost 20 years ago.
SUBMARINE CLASSES
There are several classes that U.S. submarines fall into;
SSN - SSN's are Fast-Attack submarines, and are the hunter-killer descendants of the old WWI/WWII Fleet boats. Far more lethal, they can carry a range of weapons, from cruise missiles, torpedoes, mines, and special weapons. The only real threat to an SSN is another SSN.
SSBN - SSBN's are Ballistic Missile Submarines, and their primary mission is Nuclear Deterrence. Armed with several (U.S. boats can carry up to 24) ballistic missiles, each with several MIRV's, they provide a covert and mobile deterrent that still cannot be adequately prepared for.
SSGN - SSGN's are Guided, or Cruise Missile Submarines. Many SSBN's were converted to SSGN's after the Cold War and as a result of treaty requirements. They usually carry different configurations of Tomahawk Cruise Missiles.