Concealed carry laws vary greatly from state to state, but you can pretty much put them into three categories: No CCW at all, May issue, Shall Issue, and Constitutional Carry.
Currently, there are only three areas of the country where CCW is outright illegal for a private citizen: Illinois, Wisconsin, and Washington D.C. Open carry is, theoretically, legal in Wisconsin but from what I've heard you will very likely face harassment from LE and possibly arrest for a subjective charge disturbing the peace or disorderly conduct.
May issue is common in more politically liberal states like New York, California, Maryland, and Massachussetts (just to name a few). In these states, whether or not you get a license is left up to the discretion of the issuing authority (usually the local sheriff or chief of police). The ease of getting a license in these states varies widely depending on the state in question and even the area of the state. For example, Alabama is technically a May-issue state but most people in that state seem to have little trouble obtaining a license. Massachussetts, on the other hand, is a state in which it is nearly impossible to get a license without a great deal of money, political connections, or both. Similarly, it supposedly isn't particularly difficult to get a license in Upstate New York, but getting one in New York City is a lot like getting one in Massachussetts.
Shall issue is probably the most common system in the U.S. In this system, so long as you meet all the legal requirements (clean criminal and mental history, minimum age requirements), pay any required fees, and complete any required training courses, the state must issue you a license. The finer details of the license in shall-issue states such as minimum age requirements, fees, issuing authority, legality of open carry, places where carry is prohibited (gov't building, schools, establishments that sell alcohol, etc.), training requirements, and frequency of renewal varies significantly though.
The last category is one that seems to be growing in popularity: Constitutional Carry. In this system, anyone over the age of 18 who can legally posess a firearm may carry a gun without any sort of license. Currently, only four states have adopted Constitutional Carry: Vermont, Alaska, Arizona, and Wyoming (while WY has passed the law, I don't believe it has taken effect yet). Alaska, Arizona, and Wyoming still offer licenses so that their citizens may still carry in states that have reciprocity agreements with them while Vermont does not even offer a license.
In addition to the above, some may issue and shall issue states allow open carry without a license while other require a license for either type or ban open carry alltogether. To give a more specific picture of state laws as an example, I'll describe those of my home state, Indiana:
IN is a shall issue state in which both concealed and open carry are legal so long as one has a LTCH (License To Carry a Handgun) because the license makes no distinction between open or concealed carry. Indiana makes no distinction to the type of weapon beyond "handgun" and we have no training requirement. Our minimum age to get a LTCH is 18.
Indiana does, however, offer two different types of licenses: Hunting & Target or Personal Protection. A Hunting & Target allows you to transport an unloaded handgun to and from places like shooting ranges and hunting areas (without a license, the only instance in which you may legally transport a handgun is from the place of purchase to your home although there is currently a bill in our senate to revise this). A Personal Protection license, on the other hand, allows you all the same rights as the Hunting & Target license but also allows you to carry a loaded handgun on your person or in your vehicle for the purpose of self-defense. The only disadvantage to the Personal Protection license is that it is more expensive than the Hunting & Target (the Personal Protection license is much more popular and I've never personally met anyone who opted for the Hunting & Target instead).
Licenses are also available in two different durations: four-year expiration or lifetime. While the four-year license is less than half the price of a lifetime one, the lifetime license is by far the most popular and the licensing division of the State Police has been backlogged ever since the lifetime option became available in 2006. Because of this, a person will usually have a 2-3 month wait when applying for or renewing their LTCH. When I renewed mine, my old LTCH actually expired before the new one came (I got my first one before the lifetime option was offered). However, I was told by the State Police that I could continue to carry a handgun so long as I also had my expired LTCH and the receipt of my application for renewal on my person.
The list of places where carry is legally prohibited in Indiana is relatively short and is limited to gov't buildings, schools, and commercial aircraft. While private businesses may choose to ban CCW on their premises, such bans carry no force of law beyond charges for tresspassing if you refuse to leave the business after they've asked you to. Indiana also recognizes all out-of-state licenses although visitors from Constitutional Carry states who do not have a license are prohibited from carrying here. As far as bordering states, both Kentucky and Michigan honor Indiana's LTCH while Illinois and Ohio do not.
Our LTCH is issued by the State Police, but the applications are made through the local police department or sheriff's office. According to what is printed on my license, a LTCH may be suspended or revoked for "felony arrest and/or improper or reckless use of a handgun"
Indiana does not currently have a firearms preemption law although a locality with gun laws stricter than those of the state is rare. Indiana does have the "Castle Doctrine" which states that one has no legal duty to retreat before using lethal force in self-defense so long as lethal force is legally justifiable in the situation. It is also state law in Indiana that an employer may not prohibit their employees from having a firearm in their vehicle so long as the vehicle is locked and the firearm is not in plain sight.
Indiana also has very little restriction on the types of weapons which may be owned. Indiana has no assualt weapons ban and NFA items such as machineguns and silencers are legal so long as Federal requirements are met. The only type of ammunition which is illegal in Indiana is armor-piercing handgun ammunition. Knives are also unrestricted with the exception that switchblades and "Chinese throwing stars" are illegal, carrying of knives is prohibited in certain places (pretty much the same places where carrying a gun is illegal) and people under the age of 18 are prohibited from carrying a knife with a blade longer than 6". Other types of common self-defense weapons such as batons, tazers, and pepper spray are unrestricted although I believe you must be at least 18 years old to purchase such items.