Chinese is the world's only major written language to never develop an alphabet. Chinese has been around for thousands of years, and has a very wide vocabulary of between 10,000 and 60,000 characters. The challenge, then, is how to make a dictionary that is somehow usable, without having the spelling systems that is available to dictionaries in alphabet-based languages.
There have been various methods for doing this over the years. There have been dictionaries that sorted words under topic, and various attempts through the ages to produce some type of phonetic dictionary. However, the main type of dictionary, and the one still used today, has been the type that sorted the characters by various factors inside of them. The main factor that they have been analyzed by is radical.
While Chinese does not have an alphabet, the characters are made up of smaller elements, called radicals. A radical is a smaller or altered version of a basic Chinese character, that is combined with other radicals to form a more complicated character. Radicals can represent either a phonetic or semantic value, with the typical character made up of a phonetic radical to give the sound, and a semantic to give the meaning. The number of these radicals is fairly standard, with my dictionary giving me 226. These then, are the closest Chinese comes to an "alphabet".
The way that a character would be found in a dictionary is by finding its "main" radical, and counting its strokes. A user would then use a first index in the dictionary, listed by the amount of strokes in the radical, to find a listing of every character that is under that radical. Then, the remaining strokes in the character are counted, and a second index is consulted to find a page number where the character is more fully described.
That sounds complicated, and it is. The first problem is that since Chinese characters contain two or more radicals, it is often very hard to determine which one it is going to be indexed under. When a user is counting the number of strokes, they must know the rules for stroke formation, and even then it is sometimes hard to count strokes well.
However, as with much in Chinese, the process gets quite intuitive with practice. After some use, it is easy to remember how the radicals are sorted on the first table, and counting strokes becomes a process that can be done at a glance. Also, with the advent of pinyin, most dictionaries also have an alphabetic listing that can be used easily if the user knows the pronunciation of a character.
This, of course, is just the most common type of dictionary, and the most common type of indexing method. When I was going to school in Tainan, the Cheng Gong Daxue library had an entire first floor covered with different dictionaries, including different Chinese to foreign language dictionaries, various technical dictionaries, large literary dictionaries, dialectical dictionaries, and the like. There are a large variety of dictionaries to use, and someone who spends time studying Chinese will probably have several.