Theory
You have n pieces and you want to solve them blindfolded. Before solving, choose one position and declare it the buffer, and create an algorithm Σ that swaps the buffer with another position, the target. For every position p, create an algorithm Cp that takes the piece at p to the target and does not move the buffer. You can now swap the buffer and position p by executing the conjugate Cp Σ Cp'.
To solve the puzzle, look at the piece in the buffer position, p1. The first swap will be to p1's solved position, and the piece in p1's solved position, p2, will end up in the buffer position. The next swap will be to p2's solved position, which puts the piece in p2's solved position, p3, in the buffer position. Continue like this until the buffer piece is in the buffer position (this may take no pieces at all).
If there are still positions with unsolved pieces, you must perform a cycle break. Select one unsolved position arbitrarily, q0, and swap that position with the buffer position. The piece in q0's position, q1 is now in the buffer position. Continue as before until you reach some qi = q0. By swapping the buffer position with position qi, piece q0 will be solved and the cycle break is finished. Repeat this step until all pieces are solved.
The sequence you memorize will be exactly the sequence of swaps performed in the order they are found by the above process.
On some puzzles, pure 2-swaps like above are not possible. There are several remedies.
Swap with parity
It is possible that the swap Σ has a side effect outside of the n pieces, P, where P has order 2 and Σ P = P Σ is a pure 2-swap. In this case, if the memorization has an odd length, you will have to perform P either before or after executing the swaps, which is often done as part of another step. It is important that the algorithms Cp also commute with P. A method that uses this technique is Old Pochmann.
Impure swap
The swap S may affect other positions than the buffer and the target. In this case, swaps to those positions have to be executed specially depending on the method. A method that uses this technique is M2.
Rocking 3-cycle
In this method, you declare a buffer position and a helper position h, and use a 3-cycle T that cycles the buffer to the target to the helper back to the buffer, and setup moves Cp for each position p that do not move the buffer or helper. Memorization proceeds the same, except you must memorize the sequence of positions in pairs. To execute the pair (p q) where neither p nor q are the helper, you use Cp T Cp' Cq T' Cq'. To execute (p h), you use Cp T Cp', and to execute (h q), you use Cq T' Cq'. A method that uses this technique is Orozco.
3-style
You learn a 3-cycle algorithm Tpq for every pair of positions p and q which aren't the buffer. This is advanced.
5-style
You learn a 5-cycle algorithm Ppqrs for every 4-tuple of positions p, q, r, s which aren't the buffer. Only one guy uses this.
Pieces with orientation
If the pieces can be oriented, you should think of pieces as collections of stickers, where the position of a sticker uniquely determines the orientation of a piece. You must declare a buffer sticker position, and the 2-swap S should take the buffer sticker position to the target sticker position and vice versa (with similar extensions to 3-cycles). Memorization can then proceed the same as above, but you must track the sticker on each piece that you are swapping with. You must cycle break when you reach any sticker on the buffer piece, not just the buffer sticker, and you will have to cycle break to misoriented pieces even when they are in the correct position.
Memory methods
On a cubic puzzle, there are 24 stickers per orbit, and they are usually lettered A through X. A common way to assign letters to stickers on a cube is Speffz. The letters are memorized in pairs. In audio memorization, each letter pair is turned into a single syllable, and the syllables are recalled. In sentence memorization, each letter pair is turned into a word, and the words are placed in sentences. In PAO (Person-Action-Object) memorization, three letter pair words are placed into an image where the first word, a noun (often a person), is performing the second word (a verb) to the third word (an object), and then the images can be placed in mental rooms. A database of words that can correspond to letter pairs can be found here.
Miscellaneous
It is massively helpful to pick a fixed orientation that you hold the cube in for all blindsolves, as it helps you recall which position the pieces go in. Most solvers choose the orientation of white-top green-front based on the WCA scrambling orientation, but officially generated scrambles for blind events always result in a random orientation.
In competition, the times for memorization and solving are added to produce the final time. If you want to time your blindsolves, you can use cstimer, which also generates scrambles, and set the timer to 2-phase mode to record your memorization time and solving time separately. If you just want to generate scrambles without using a timer, you can use scramble.cubing.net.
Specific guides
3×3×3