The easiest way to explain is this: 

Given a circle with radius 1 (the so called 'unit-circle'). We take a random point on this circle. We can describe this point uniquely by the angle of the radius through this point and the X-axis, for example 90, 140, ... 

Mathematically however, radians are used instead of degrees. Without going into much theory, you should just remember that 360 = 2*PI. 
So suppose we have a point lying at N radians on our unit circle. Then the X coordinate will be given by cos(N), and the Y coordinate by sin(N). That's the definition of cosine and sine. 
So if you want to draw a circle, take every value A from 0 to 2*PI, and plot the points (cos(A), sin(A)). Easy. 
The relationship of sine and cosing with right-angled triangles is pretty easy to see. Draw a circle using the origin (0,0) as center. Take a point on this circle, connect the point with the center, project the point on the X axis. Voil: a right-angled triangle appears  

Sin and cos have other uses as well. Because they are coordinates of points lying on a circle, they are periodic. A point traveling along a circle will eventually arrive at its starting point. This means that their values are repetive after a while (2*PI to be exact). 

So you can use sine and cosine as a source for a pulsing/wavy scope or movement. 

The tangent tan is defined as sin divided by cos. It ranges from negative infinity to positive infinity in -PI/2 to PI/2, and repeates itself every PI. 

Usually you won't need the actual mathematical uses of these functions, but you'll rather be using their characteristics (e.g. repetiveness).