Which one should I use: begin{align} or begin{aligned}?

If you have a series of equations that you want to have mutual alignment, then you would use align. If the equations are part of a larger equation, then aligned is the right environment.

I often will use begin{equation}begin{aligned} ... end{aligned}end{equation} to give a single equation number to a group, for example if I'm showing steps.

align is used for entering to math-mode while aligned is used for multiple horizon alignment. For example

 begin{align}
 begin{aligned}
    2x+3 &= 7 & 2x+3-3 &= 7-3 
    2x &= 4 & frac{2x}2 &= frac42
    x &= 2
 end{aligned}
 end{align}

ps. the example is adapted from Math Mode document of Herbert Voß.

For grouping the multiple lines equation, you should use gather or multline.

You can use align inside subequation to get the numbering (1a) and (1b). Then you can refer the the equation system with (1) and to the subequations directly.

begin{subequations}  
begin{align}  
a &= b   
c &= d  
end{align}  
end{subequations}