Pythagorean Triples
A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:
a^{2} + b^{2} = c^{2}
Example: The smallest Pythagorean Triple is 3, 4 and 5.
Let's check it:
3^{2} + 4^{2} = 5^{2}
Calculating this becomes:
9 + 16 = 25
And that is true
Triangles
And when you make a triangle with sides a, b and c it will be a right angled triangle (see Pythagoras' Theorem for more details):
Note:

Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:
Here are some more examples:
5, 12, 13  9, 40, 41 
5^{2} + 12^{2} = 13^{2}  9^{2} + 40^{2} = 41^{2} 
25 + 144 = 169  (try it yourself) 
And each triangle has a right angle!
List of the First Few
Here is a list of the first few Pythagorean Triples (not including "scaled up" versions mentioned below):
(3,4,5)  (5,12,13)  (7,24,25)  (8,15,17)  (9,40,41) 
(11,60,61)  (12,35,37)  (13,84,85)  (15,112,113)  (16,63,65) 
(17,144,145)  (19,180,181)  (20,21,29)  (20,99,101)  (21,220,221) 
(23,264,265)  (24,143,145)  (25,312,313)  (27,364,365)  (28,45,53) 
(28,195,197)  (29,420,421)  (31,480,481)  (32,255,257)  (33,56,65) 
(33,544,545)  (35,612,613)  (36,77,85)  (36,323,325)  (37,684,685) 
... infinitely many more ... 
Scale Them Up
The simplest way to create further Pythagorean Triples is to scale up a set of triples.
Example: scale 3,4,5 by 2 gives 6,8,10
Which also fits the formula a^{2} + b^{2} = c^{2}:
6^{2} + 8^{2} = 10^{2}
36 + 64 = 100
If you want to know more about them read Pythagorean Triples  Advanced