How does one align a laser using two adjustable mirrors? Suppose we would like to get a laser beam to pass through a particular point in space in a particular direction.
At our disposal we have a laser, two adjustable mirrors (can rotate up/down and left/right) and a card which we can hold in the laser beam to see where it is.
What is the procedure we should follow for adjusting the positions and orientations of the mirrors to achieve the goal?
How, mathematically, can we see that this procedure converges (i.e. gets the laser aligned within a reasonable number of steps)?
 A: I would start by taking a pen and a piece of paper and drawing the relative positions of the source point and the target. Then draw the direction of the laser beam as a long and a straight line starting at the source point, lets call this Line1. Try to draw a line parallel to the laser beam that goes through the target point (Line2). Now, draw a new line (Line3) that is perpendicular to both Line1 and Line2. This Line3 should intersect with Line1 and Line2. You will place the mirrors on these intersections points of Line1 and Line3 or Line2 and Line3 respectively with approximately 45 degrees to the incident and reflected beams. Planning part is over. 
Now for the actual aligning part: generally, the laser and the mirrors are placed on an optical bench, that has equally spaced holes on it. You can use those holes to see if you beam is going straight. To do that, use the card you have. Hold the card so that you can see the beam, try to look at the dot on the card straight from the top and see if it coincides with our imaginary line created by the holes. 
Place the mirrors approximately at the positions you planned with ~45 degrees with respect to the incident beam (actually as long as both of the incident angles of the mirrors are a total of 90 degrees it is fine). I would set the height of the mirrors so that the centre of the mirror is approximately at the same height as the laser aperture and the centre of the second mirror with the target point. We want to stay away from the edges of the mirrors. This ensures that in the end, we will have a beam that goes straight into the target without any tilts. 
Now, with the card, make sure that the beam is centred on the first mirror, if not, change the position of the mirror slightly to make it so by moving the post holder that holds the post on which the mirror1 resides. After it is centred on mirror1 and you fixed the post holder of mirror1, you can change the lateral angle of the mirror1 so that the beam is incident on the same vertical line as the center of mirror2 by rotating the post of mirror1 while its post holder stays fixed. We did not use the knobs yet, mind you. Now try to find the knob that will move the beam to ~center of mirror2 in the vertical direction. After that is done too, fix the post holder of mirror 2. 
Now turn the post of the mirror2 in lateral direction inside the fixed post holder so that the beam is on the same vertical line as the target point. Then turn the knob so that the beam moves in the vertical line and is on top of the target. You have aligned the laser to the target.
Now lets say you went out for a quick bite and when you return you see that the beam is not aligned anymore (it happens). Now do not move the post holders of mirror 1 and mirror 2 that you fixed before. You can simply use the knobs as the misalignment cannot be too bad afterall. mirror1 is far away from the target with respect to mirror2, so rotating the knobs on mirror1 will move the beam much farther than the knobs of mirror2. 
This may seem like a very long procedure, but it really is not. When you practice it 3 or 4 times, you can do this under two minutes. 
Also note that this is the procedure for aligning the laser to a single target point. The procedure to align the laser to the optical rail is not the same although not entirely different.  
A: Set two points, one as far away as you can get, and one close to the 2nd mirror. 
Use mirror 1 to align for the close point and mirror 2 to align for the far point. Keep iterating over this (M1 -> M2 -> M1 -> M2) until the beam goes through both points at the same time. 
