3-2-3 edge pairing
3-2-3 edge pairing is a technique to pair the last eight dedges on 4x4x4 and up in pairs of 3, 2 and 3.
This is accomplished by placing the four already solved dedges on the bottom face (in most methods like Yau or Hoya, this is already done in the Cross step). After that, a Uw (or Uw') move is performed to misalign the centers. Then, the solver solves three edges by looking at the edge on the top left of the E-slice and placing the corresponding edge in the bottom right, so that they would pair if a Uw' (or Uw) were to be performed. This is repeated three times and followed by a Uw' (or Uw) to pair up the three edges. The next to edges are solved via the "standard/beginner way", which is to place two edges in the E-slice, pair them with a wide U and insert another edge so that two edges are paired in the end. For the remaining three, this is repeated again (since it is impossible for only one dedge to be incorrectly paired on 4x4x4, the last step pairs up one more dedge than the one before it).
Although method of edge pairing is the most popular one among 4x4x4 solvers, especially for Yau, it is usually not used for 5x5x5 and big cubes.
4-2-3 edge pairing
4-2-3 edge pairing is a technique to pair the last nine dedges on 4x4x4 and up in pairs of 4, 2 and 3.
4-2-3 is done almost identically to 3-2-3 edge pairing, except that the first slice solves an edge. After slicing back, 3 more are solved. This is followed by another 3 edges and the final 2, thus 4-2-3. While mainly used on 4x4x4 for HK+, this can also be applied to 5x5x5 by first solving the outer wings using 4-2-3, and then solving the inner ones using 4-2-3 with slices.
6-2 edge pairing
6-2 edge pairing is a technique to pair the last eight dedges on 4x4x4 and up in pairs of 6 and 2.
Like in 3-2-3 edge pairing, the four paired dedges are placed in D and the solver puts three edges relative to each other so that they would be paired by a Uw (or Uw'). However, no wide U move is performed before placing the edges. To create three dedges, the Uw (or Uw') is performed. This is followed by pairing three edges again the same way (although now, the centers are misaligned by a Uw' (or Uw)). To finish pairing the six dedges, the Uw' (or Uw) move is performed. Now to finish the last two dedges, the edge in the U layer is put diagonally from the other unsolved edge in the E-slice (e.g. one in BL and one in FR position) so that the top and bottom of both edges match (i.e. one dedge won't be paired by a Uw2). To pair both edges at the same time, the solver performs a Uw2, a flipping algorithm and another Uw2.
6-2 edge pairing is the second most common edge pairing technique among 4x4x4 solvers (after 3-2-3) and often used in conjunction with Hoya (although it can also be used with any other method that pairs up four edges beforehand, e.g. Yau). Similarly to 3-2-3, it can also be applied to 5x5x5+ but isn't really used by anyone.
Freeslice edge pairing
Freeslice edge pairing is a technique used on big cubes ranging from 4x4x4 and up in which the slices between the U and D layers do not have to be re-solved immediately after an edge pair is formed. Rather, the slices are re-solved after 8 edges (or only 4 if 4 other ones have already been created like in Yau or Hoya) are paired and when there are only 4 edges are remaining to be paired.
Essentially all world-class big cube solvers use this technique for 5x5x5+ to some extent and it is also the most common one for puzzles higher than 4x4x4. When this is combined with the Reduction Method, the result is usually referred to as the Freeslice Method.
AvG edge pairing
AvG Edge Pairing (named after Arnaud van Galen), also called '2 edges at a time pairing, is an edge pairing method for the 5x5. Notable users of it are Arnaud van Galen, Erik Akkersdijk and Durben Virtucio.
Big Cubes edge pairing
Big Cubes edge pairing is an edge pairing method for the 5x5 and bigger cubes, popularized by BIGcubes.com. Together with AvG edge pairing, it is the main edge pairing method for 5x5. For bigger cubes, Big Cubes edge pairing is by far the most popular method among top cubers. Notable users of it are Dan Cohen, Kevin Hays, Feliks Zemdegs, Mats Valk, Max Park and Seung Hyuk Nahm.
UEF8E is a method for pairing the first eight edges on big cubes. By using U2, the solver can avoid cube rotation and using flipping algorithms like F' L F U L2 U'. While it reduces the move set (<L, Lw, 3Lw, U, R>-gen on 6x6x6 and <L, Lw, 3Lw, 4Lw, U, R>-gen on 7x7x7), the solver has to memorize four useless wings.