This is the first in a new series on this blog called "how do you say", in which I take a word in English that has multiple translations in a given target language and try to figure out which ones are used for what. As I'm doing this to answer my own questions (or maybe yours—leave them in the comments!), drop a line in the comments if you think I've gotten someone wrong and I'll revise as necessary.
When you look up the Japanese for the English noun "pattern" on ALC, you get (deep breath, now) 傾向 keikou, 模様 moyou, 型 kata, 原型 genkei, 型紙 katagami, 柄 gara, 形態 keitai, 構図 kouzu, 模範 mohan, and パターン pata-n (as if they didn't have enough ways to say it with Chinese characters, they had to bring in a loanword from English as well). Let's see if we can figure out what to use when.
Thanks to that loanword, I've got good news for you for you lazy native-English speakers. The basic rule appears to be that, when in doubt, use パターン pata-n in good old katakana, taken straight from English. パターン pata-n has 18 pages of examples where the English equivalent is "pattern" on ALC. Of all the other ways to say "pattern" in Japanese listed above, the only ones with more than one page of such examples are 型 kata (with two) and 模様 moyou (with three). What's more, looking through the numerous examples of パターン pata-n on ALC, there appear to be relatively few cases where some construction of パターン pata-n can't replace the other words below.
Turning to the the one with the second-largest number of such examples on ALC, 模様 moyou means "pattern" in the sense of a print or weave design (e.g., 市松模様 ichimatsu moyou, "checkered pattern").
Also in the two-dimensional realm, 柄 gara is used to mean patterns on cloth and certain similar things (wallpaper, etc.). The difference between this and 模様 moyou seems to be that, for 柄 gara, the pattern is generally part of the material rather than just printed on it (e.g., アーガイル柄靴下 a-gairu kutushita, "argyle socks"). However, this appears to be a rule of thumb rather than a hard-and-fast rule (e.g., ヒョウ柄 hyou gara, "leopard print pattern").
型 kata seems to be the Japanese translation of "pattern" with the broadest use. For those language learners among us (can I see some hands?), this is the term to use for a linguistic pattern (e.g., 動詞型 doushigata, "verb patterns"). And there appears to be a wide variety of other uses. I'll refer you to ALC and if you can clarify this for me any better, drop a line in the comments below.
The related 型紙 katagami means a pattern made on paper, such as a cut-paper stencil or a sewing pattern.
模範 mohan means "pattern" in the sense of an example that is to be emulated. The word has a good connotation, e.g.: ～を模範とする ~ wo mohan to suru, "to pattern oneself after"; and 親は、子の模範 oya ha, ko no mohan, "Parents are patterns for their children".
傾向 keikou means "pattern" in the sense of "trend" or "tendency". So if you're talking about patterns of behavior over time (e.g., 消費傾向 shouhi keikou, "spending pattern"), you can use this one.
形態 keitai seems to be used for patterns of human activity that can be characterized at a given point in time or based on data collected, in contrast to 傾向 keikou, which considers ongoing activity and the future. Examples include: 人口形態 jinkou keitai, "population pattern"; 労働形態 roudou keitai, "working patterns"; 暴力形態 bouryoku keitai, "pattern of violence"; etc.
Although 原型 genkei comes up under pattern (e.g., 原型製作 genkei seisaku, "pattern manufacture"), it means the original upon which subsequent things are patterned after, so "prototype", "original", "model", etc., tend to be better translations (e.g., 原型製作 genkei seisaku can also be translated as "prototype production").
構図 kouzu appears to mean "pattern" in the sense of structured relationships (e.g., 政・官の癒着の構図 sei ˙ kan no yuchaku no kouzu, "pattern of collusion between politicians and bureaucrats"). It does not appear to be a very common way to translate "pattern".
So... anything wrong here? Anything to add? Drop a line in the comments below (especially you native Japanese speakers out there!).