I'm only going to touch on the major and minor scales in this post, because they are the most commonly used scales in modern music, and once you have an understanding of how scales fit together, you can play them anywhere on the neck.
There are two common ways to play a major scale:
e:|-|B|C|-|-|-|
B:|-|-|G|-|A|-|
G:|-|D|-|E|F|-|
D:|-|A|-|B|C|-|
A:|-|E|F|-|G|-|
E:|-|-|C|-|D|-|
And:
e:|-|-|D|-|E|F|
B:|-|-|A|-|B|C|
G:|-|E|F|-|G|-|
D:|-|B|C|-|D|-|
A:|F|-|G|-|A|-|
E:|C|-|D|-|E|-|
You'll notice, as you travel up the strings, that even though the patterns are different, the notes occur in the same sequence. This is because the same note occurs in multiple places on the neck, and can be played in different places to suit different fingerings and even styles of play. The top example is the more traditional way of playing a major scale, and lends its self more easily to melodic playing. The bottom is often referred to as a "shredder scale", because there are three notes per string and, as you can see, small modules of notes on the E&A, D&G, and B&e strings. This makes it easy to blast up and down your scales at silly speeds, and is usually favored by metal guys.
We can see the same thing with the minor scale:
e:|-|A|-|-|-|-|
B:|-|E|F|-|G|-|
G:|B|C|-|D|-|-|
D:|-|G|-|A|-|-|
A:|-|D|-|E|F|-|
E:|-|A|-|B|C|-|
And:
e:|-|-|-|-|-|-|
B:|-|F|-|G|-|A|
G:|C|-|D|-|E|-|
D:|G|-|A|-|B|-|
A:|D|-|E|F|-|-|
E:|A|-|B|C|-|-|
The really important thing to learn here is that the same note occurs one string down and five frets to the right. This is important, because if you're confronted with a situation where you've run out of fingers and can't move any further to the right on a string, you can always step up to the next string using this rule.
A:|X|-|-|-|-|-|
E:|-|-|-|-|-|X|
Octaves occur two strings up and two frets to the right. This is one of my favorite tricks, because it lets you repeat the same riff in different octaves on the neck. It sounds very shreddy, and is a useful way of moving from place to place quickly.
D:|-|-|-|X|-|
A:|-|-|-|-|-|
E:|-|X|-|-|-|
This next example is one such riff. The X'es mark the octaves, and the 0's mark the other notes. You can see how it repeats up the fretboard. If you look closely, you may notice that my rule does not hold up between the octaves on the D and B strings. This is because the distance between the G and the B strings is one half-step shorter than all the others, which equates to one fret shorter. One half-step equals one fret.
e:|-|-|-|-|-|-|0|-|-|0|-|-|
B:|-|-|-|-|-|-|X|0|-|-|-|-|
G:|-|-|-|0|-|-|0|-|-|-|-|-|
D:|-|-|-|X|0|-|-|-|-|-|-|-|
A:|-|0|-|-|0|-|-|-|-|-|-|-|
E:|-|X|0|-|-|-|-|-|-|-|-|-|
Because of this, every time we play up to the B string from the G string, we need to shift the next note one to the right. Similarly, move one to the left when traveling downwards.
e:|-|-|-|-|-|-|0|-|-|0|
B:|-|-|-|-|-|-|X|0|-|-| --> Traveling upwards; move right.
G:|-|-|-|0|-|-|0|-|-|-| <-- Traveling downwards; move left.
D:|-|-|-|X|0|-|-|-|-|-|
A:|-|0|-|-|0|-|-|-|-|-|
E:|-|X|0|-|-|-|-|-|-|-|
Here's an example: On any other two strings, the beginning of a C major scale will look like this:
A:|-|F|-|G|-|A|-|
E:|-|C|-|D|-|E|-|
But on the G and B string, it will look like this:
B:|-|-|F|-|G|-|A|
G:|-|C|-|D|-|E|-|
The reason for this is that it makes barre chords work. With the B string down a step, it would be impossible to barre chords.
