The first formulation of the Categorical Imperative is this: "Act only in accordance with that maxim through which you can at the same time will that it become a universal law". To perform this test, you must first picture the action you are about to take as a maxim. According to Kant, a maxim takes this form: "I, a person in situation Z, will perform action X in order to bring about Y". After doing so, imagine that your maxim is a universal law, that is imagine that every rational agent who is in situation Z and acts with Y as their end must inevitably seek it by doing X, as though it were a law of nature.
There are two ways that your action may fail this test. It may be that you can't will that your maxim be universal law (because universalizing it leads to a logical contradiction, and is thus impossible to will that maxim to be a universal law) or it may be that you can't will that your maxim be universal law (universalizing it is logically possible, but the world that results from it is not one you could rationally will to live in).
An example of the first case: Suppose that you are in need of money, and you decide to lie to someone to borrow money from them with no intention of paying back. Now imagine a world where this action became a universal law: Every time you went to do that, the persons who are in a position to lend you money will be aware of what you are doing (because everyone always does that) and will refuse to be manipulated by you, refusing to lend you the money, thus preventing you from obtaining your willed end Y. Because you failed to obtain your willed end, you cannot successfully act according to your maxim. This is a logical contradiction: You cannot possibly will that your maxim become universal law, because once universalized you can't even possibly act according to your maxim at all. Kant would argue that you have a perfect duty to be honest in your affairs and refrain from manipulating people into giving your their money when they otherwise wouldn't if you spoke the truth.
An example of the second case: Suppose another persons are in need of assistance, and in order to avoid having to making an effort, you decide to never assist or aid anyone else. Now imagine this maxim were a universal law: The world where no one ever helps anyone else is logically possible, but it would be a very terrible world wouldn't it? Life would be nasty, brutish and short to everyone. Can you rationally will to live in a such a world? I think not, thus you cannot will this maxim to be universal law and it fails the test. Kant would argue that you have an imperfect duty to assist and aid other persons.
The difference between perfect and imperfect duties is that perfect duties are something you are supposed to do all the time with no exception, whereas imperfect duties are things you have to do but you can allow your inclinations and circumstances to influence when and to what extent you are going to do it. Simply put, if you will to live in a world where people assist one another then you have a duty to help bring this world about by assisting and aiding other people yourself, but you don't need to be assisting and aiding people full time you just have to do it sometimes and to some extent.
If your maxim passes the test - your maxim is both logically conceivable and rationally desirable once universalized - then according to Kant your action is morally permissible.
As you can see, the second way that this test can fail is similar to the Golden Rule (though it's still somewhat different from it). The reason why the Categorical Imperative is "stronger" than the Golden Rule is that it has one more "layer" of moral testing, so to speak. If one acts solely according to the Golden Rule, a person who enjoys feeling pain would be able to justify when they coercively inflict pain upon others by saying they are just doing unto others what they would have them do unto themselves - but if one acts according to the Categorical Imperative, before saying they do indeed wish to live in a world where people forcefully inflict pain on one another all the time this person who enjoys pain would first have to prove this world to be logically coherent, which would not be the case. So the Categorical Imperative forbids plenty of things which the Golden Rule would allow.
This is but the first formulation of the Categorical Imperative, there are three other formulations (Kant held all the different formulations to be equivalent), which you may find more intuitive and easier to actually apply. For example, the second formulation: "Never act in such a way that we treat humanity, whether in ourselves or in others, as a mere means only but always as an end in itself." What this formulation means is that you ought to recognize humanity itself as absolutely valuable, and respect both yourself and other people as valuable rational agents who have their own aims and desires and are free to make their own choices and are responsible for them. When you manipulate people or try to force them to pursue your own private ends against their will, you are disrespecting their humanity and failing to act according to the Categorical Imperative. In a very simple nutshell this formulation means "Don't use people". It also applies to yourself: If you take actions that cause lasting damage to your humanity in order to fulfill some some private aim that is not as valuable as human dignity itself (say, by becoming addicted to dangerous drugs or by selling an organ just to make some quick cash) you are using yourself as a mere means to an end and this violates your dignity as a human being and is thus immoral.
Sources: Kant's Moral Philosophy on SEP.
The first chapter of Christine Korsgaard's "Creating the Kingdom of Ends".