(I'd like to preface this article by saying this method has been really useful for me, but if you are starting from zero then you should probably pick up your first few kanji through Genki or a kanji workbook for beginner's. If not, try using this method while going in order through the first part of Heisig's Remembering the Kanji. This method relies on learning lots of primitives, which is confusing if you are completely new to kanji. Once you've learned a few kanji you will start to figure them out very quickly, and this method becomes much easier. It doesn't take long, so don't worry!)
Let's take a quick look at kanji: What they are, how they work, and how to learn them. It may be helpful to install Rikaikun / Rikaichan when learning kanji - it's like a popup dictionary. If you want, take a second to install it now.
To do this let's examine one of my favorite words: 花火・はなび・Hanabi・Fireworks.
First off, a history lesson! Kanji (漢字) basically means "Chinese characters" - and that's exactly what they are. A long time ago, Japanese didn't have a writing system. Over time with increasing interaction between the Chinese states and Japan, these characters were brought over to Japan. From that point, they were assigned to Japanese words and sounds.
This means that the basis of the Japanese language is not based on Chinese - only the characters used for writing. (Hiragana and katakana, by the way, are actually super-simplified forms of kanji.) This adaptation has led to many strange quirks, but we can deal with those later.
花火 is a compound word made of two kanji. The one on the right is simpler, so let's look at it first.
火 - We can see a bit of info on this character if we look it up in a dictionary, like here. If we look at this page, we see that this is a common kanji with 4 strokes, and it generally has a meaning like "fire."
If you look further down, you can see the stroke order - basically, how it should be written. Try to get very familiar with stroke order. Just as with Latin characters, knowing the movements the pen takes to create a character is almost more important than the specific form it takes in a particular font. Knowing the stroke order will help you to read various fonts, handwriting, and most people agree it helps you to remember the character as well.
You can also see what Jisho calls "Japanese kun." This is in reference to 訓読み・くんよみ・Kun'yomi - or Japanese Reading. Next is "Japanese on" which refers to 音読み・おんよみ・On'yomi, or Chinese Reading. It's important to note that Chinese Readings in Japanese are not equivalent to the modern reading of the same kanji in Chinese. Sometimes the On'yomi represents a reading as it existed in China long, long ago - and sometimes it represents a sound that Chinese never actually used!
In general, Kun'yomi is used when a kanji is used by itself to create a word and On'yomi is used when more than one kanji is used to create a word. There are exceptions to this - 花火 being one of them. Some readings are very common and others are very rare. It's almost useless to try and remember readings by themselves, though. You should learn vocabulary with new kanji instead, in order to get familiar with common readings. You can find common words with Beta Jisho by doing a search like this.
Here we can see that the word for "Tuesday" is 火曜日. You might want to write that down in your notes, or you could look for something that seems a little easier - maybe 火山 ("Volcano"). From these, you can see that the カ reading is very common.
But now you need a way to remember fire. Sure, it's easy and looks like fire, but this is a very simple kanji. That 曜 in 火曜日 isn't quite as easy, for example. So let's learn one way to use mnemonics now.
If you pick up a copy of Heisig's Remembering the Kanji he will introduce you to a number of good mnemonics. The Heisig Method involves breaking kanji down to simple primitives - parts that are frequently repeated in various kanji. They don't have a meaning by themselves (and really, most kanji don't have a specific meaning until they're put in a word), but we give them names to help us remember them.
火 is very simple, so it is used as a kanji without any primitives. In fact, 火 itself will be a primitive in later kanji! It's very common for simple kanji to be used inside of more complex kanji.
So we need a story for this kanji. We can use Heisig's from his book, but sometimes his stories don't stick for me. Or we could make our own, but we don't have much experience using good mnemonics yet and it can take a long time to come up with mnemonics for 2000 kanji. So I recommend going to this site, signing up for a free account, and using the mnemonics that other people have created!
When we look on the Koohii page for 火, we can see that it is kanji #161, has a reading of カ, a name of FIRE, and there are many mnemonics. Remember, kanji often have very general meanings or even more than one meaning! The name we use for a kanji is just a helpful reminder. It's the vocab that will really help you to understand the ideas behind a kanji.
So we look at one of the top mnemonics and pick one that seems helpful for us, then click the "star" image to save it.
For example, user 'munia' has written:
For some reason, this kanji reminds me of a person running, her arms (the two drops) raised up, because she caught on fire!
If I'm keeping a kanji notebook, maybe I've written down that kanji nice and large, the name of the kanji, one or two common readings, one or two common vocabulary, the primitives (remember, 火 doesn't really have any), and now I will write down that mnemonic - which I will try to remember whenever I see the kanji in the future.
Great, one kanji down. I'll look over my notebook periodically to quiz myself and refresh my memory, and I'll look to it if I encounter 火 in the wild and forget its meaning. Let's move on to 花.
If we put that guy into a dictionary, we see that it means "flower." If you see this written and can't copy-paste it on a computer, you can try a radical search. If you hit the button to the left of the search box on Beta Jisho it let's you do a radical search. Here you pick the radicals ("official" primitives) that are in the kanji to try and find it. The radicals are sorted by the number of strokes involved. So in 花 we have three radicals, two on the bottom with two strokes and one at the top with three.
Okay, if we look up the kanji details we can see the same kinds of information that we did with 火. Take your notes, find some simple and common vocabulary (花 by itself is はな meaning flower, which is a good one to demonstrate the Kun'yomi!), etc.
But wait! 花 is read はな, カ, or ケ! Isn't 花火 はなび? Yes it is, there's always exceptions to the rules and one-off readings. Also don't forget that sounds can be changed when but into a compound, so while 日 is ひ it is changed to び in 火曜日 to make it easier to say.
Now let's look up a mnemonic. This kanji has primitives! Let's see how they work in remembering kanji.
Uh-oh, we have a few problems. If you are going through the book Remembering the Kanji you won't have these problems, and maybe that's a good idea. But we can also work around it!
In mnemonics, many people like to italicize the names of primitives that they implement into their story. We can gather from these mnemonics that the primitives in 花 are flower (confusing, but useful) and change. But which part is which?!? If you have Remembering the Kanji you can flip through and find the primitives. Or, you can do a google search for "Flower Primitive" or "Flower Primitive RtK," something like that.
Or you can try typing the name of the primitive into Koohii's search box to see if you can find it. Luckily, we can find change! It's the bottom part of the kanji in 花, 化! So maybe you want to add that kanji to your notebook before finishing the 花 entry, or maybe not. You can do it however you like, as long as it is helping you remember these kanji. I find it useful to make sure that I always understand the primitives, so I would add 化 to my book.
And I'll just give you the other radical/primitive so you don't have to look it up: It's the three strokes at the top of 花, it is called "flower" in the Heisig method (because it is a part of the kanji for flower, and because it kind of looks like little flowers popping up through the ground), and we use it in lots of kanji.
So now we have the kanji, readings, vocabulary, and primitives written down. We're ready for a mnemonic!
Maybe something very visual like "Imagine the Kanji primitive for flower miraculously changing into the full Kanji for flower right before your eyes" will help you. Or maybe something like "The flowers change with the seasons" is better. It's up to you!
Okay, now you understand a bit about what these kanji mean, how they are read in a few words, and you have a trick to help you remember them next time you see one. You have the basics nailed down. The rest is learning more vocabulary, consuming more material and encountering the kanji more, and maybe putting things into an Anki deck to help quiz yourself! If you do flashcards, do them from English -> Japanese. And remember, there are no concrete rules in Japanese kanji. There are so many exceptions, special uses, idiosyncrasies, and so on. Remain calm and have faith that things will make a whole lot more sense with just a bit more studying. =)