What methods are available for looking up a character?
This question could probably best be answered by Wikipedia as there are many, many methods.

One relatively common one is to look up the character by stroke count, then by stroke order. In this system, there are five types of strokes and each is assigned a number.

+--------+------------------------------+
| Number | Stroke type                  |
+--------+------------------------------+
|      1 | Horizontal, or rising        |
|      2 | Vertical                     |
|      3 | Falling to the Left          |
|      4 | Dot, or Falling to the Right |
|      5 | Turning                      |
+--------+------------------------------+


This is the method used to look up characters in 现代汉语通用字笔顺规范, the official character stroke order reference in mainland China. For example, below is the character 永 in this book:



If you want to look up this character, you will first go to the page with the 5-strokes characters, then you would search for the dot stroke (4), then the hook (5), etc. Basically, its code would be 45534.