Those crazy things that music conductors make, waving those little sticks around on that little box they stand on.

The basics. A conductor holds the baton in his right hand, which is the "beat" hand. This hand is used to indicate the beats in a measure, the tempo, the style, dynamic, and general expression the conductor wishes his orchestra/chorus/band to play. The left hand is traditionally used for cueing, that is, pointing to or giving an upbeat to a specific section that is coming in from a rest.

Time Signature Patterns:
• 2/4 - 2/4, which means 2 beats to a measure where a quarter note is one beat, is one of the simplest patterns to conduct. As with all conducting patterns, the first beat is down. In 2/4, the pattern looks like a subtle J shape, where the 1st beat is at one end of the J (going downwards) then the 2nd beat is at the top of the J. The "curve" at the bottom of the J is formed by a wrist movement, not an arm movement, to accent the 2nd beat.
(2)
|
|
|
|
\__(1)
• 3/4 - 3 beats to a measure where a quarter note is one beat. In 3/4, a right triangle-ish shape is made. The first beat is downwards, as always, then the next beat is a sweeping arc to the left, then the third beat is back up to the top of the pattern. Each beat is accented and "curved" with a wrist movement. For simplicity's sake (and my poor ASCII skills), this is the pattern without the wrist movement.
(3)
|  \
|   \
|    \
(1)---(2)

• 4/4 - By far the most common time signature, also called "common time" in some musical notation. It means (you guessed it) 4 beats to a measure, where a quarter note is one beat. The 4/4 pattern is the same as the 3/4 pattern, excepting that a movement to the left is added for the 2nd beat. The pattern, shown without wrist movement:
(4)
|  \
|   \
|    \
(2)---(1)---(3)

• Etc etc... - From there on, the patterns tend to just get bigger and more complicated. While there are many pieces written in 6/8, 8/8, 12/8, etc, most conductors will choose to "reduce" them and conduct in a smaller pattern. However, for conducting patterns of numbers higher 4, here's a few extremely crude examples (as usual, without necessary wrist movement drawn in). These will give you an idea, but for the love of God, please find someone to demonstrate them for you before you run out to conduct in one, or really any, of these patterns.
5/4            (5)
|  \
|   \
|    \
(3)---(2)---(1)---(4)

In 5/4, the distinction between beats 1, 2, and 3 is made by making a little "jump" to each beat, and accenting the beat with a wrist flick. There are two 5/4 patterns, one being with the 3 on the left, and one being with 3 on the right. The difference is in which beat you are accenting (just like 3/4 sounds weird unless you accent beat 1, 5/4 sounds weird without either beat 3 or 4 accented). The above pattern is with 4 accented.

For 6, 7, 8, etc beats in a measure, just add beats on either side according to the template of the 5/4 pattern.

Stylistic Patterns. These are, of course, a lot more about you than what the book says (no one's going to think you're a bad conductor if you have a different marcato pattern than their last conductor. If you flub 3/4, on the other hand...) Basically, it's all about just feeling the music. When the dynamics get louder, the pattern gets bigger. When the music is soft, the pattern gets smaller. When the notes are staccato or marcato, a harder accent and stop between beats is used, when it's legato a more flowing motion is used. It's all pretty much self-explanatory.

This is, by far, not all there is to know about conducting patterns. Hundreds of books are available on the subject, you can learn yourself into oblivion on the subject in college, and just asking a seasoned conductor will get you lots of information. These are just basic techniques for the beginning student.