According to Columbia Electronic Encyklopedia : Montage (montäzh', Fr. môNtäzh'), the art and technique of motion-picture editing in which contrasting shots or sequences are used to effect emotional or intellectual responses. It was developed creatively after 1925 by the Russian Sergei Eisenstein; since that time montage has become an increasingly complex and inventive way of extending the imaginative possibilities of film art. In still photography a composite picture, made by combining several prints, or parts of prints, and then rephotographing them as a whole, is often called a montage or a photomontage. [ABSTRACT FROM PUBLISHER]

Sergei Eisenstein is a father of the montage of attractions. In 1923 he explain in his essay that: An attraction (in our diagnosis of theatre) is any aggressive moment in theatre, i.e. any element of it that subjects the audience to emotional or psychological influence, verified by experience and mathematically calculated to produce specific emotional shocks in the spectator in their proper order within the whole. These shocks provide the only opportunity of perceiving the ideological aspect of what is being shown, the final ideological conclusion. (Eisenstein, 2009a, p.30)

Montage is a combination of the compositional generalization about the image and the image itself: a purposeful ‘fusion’ of compositional elements together with a generalized ‘contour’ of the image.(Eisenstein, 2010, p.4) Shot was broken into fragments. The juxtaposition of two details produces a representation of another, the psychological. For example water and eye signifies ‘to weep’, representation of an ear next to a door means ’to listen’, mouth and bird signifies ‘to sing’.

In 1929 in his essay ‘Beyond the Shot’ Eisenstein explained his theory of montage of attraction by comparing a montage to Japanese hieroglyphs. He pointed that two symbols put together make a new meaning. The combination of two hieroglyphs of the simplest series is regarded not as their sum total but as their product, i.e. as a value of another dimension, another degree: each taken separately corresponds to an object but their combination corresponds to a concept. The combination of two ‘representable’ object achieves the representation of something that cannot be graphically represented. (Eisenstein, 2009c, p.82) Here is brilliant example for this type of montage:

In this sequence Eisenstein used the comparison of subjects for thematic effect as a method of montage of attraction. In 'The Montage of Film Attraction' he explained that he uses this kind of montage to avoid overacting among the extras from the labour exchange 'in the business of dying' but mainly to excise from such a serious scene the falseness that the screen will not tolerate but that is unavoidable in even the most brilliant death scene and , on the other hand, to extract the maximum effect of bloody horror(Eisenstein 2009b, p.38)

Eisenstein pointed also that the shot is by all means a montage element. It is a montage cell. And what characterizes the shot is a conflict between two neighboring fragments. He distinguished the conflict of:
graphic direction (lines),
shot levels (between one another),
spaces etc.,
close-ups and long shots,
different directions of graphic symbols,
conflict between light fragment and dark.
Conflict should appear not only within the shot but also within the frame. In his essay 'The Dramaturgy of Film Form (The Dialectical Approach to Film Form)' he described montage as not an idea composed of successive shots stuck together but an idea that DRIVES from the collision between two shots that are independent of one another. (Eisenstein, 2009d, p.95)

In 1929 Eisenstein wrote 'The Fourth Dimension in Cinema'. In this essay he explained what is an orthodox montage. According to him it is a montage by dominants. The combination of shots according to their predominant (principal sign). (2009e, p.111) Montage by tempo, by duration of sequences, by foreground. The dominant signs of two shots side by side result in a particular conflict relationship that produces a particular expressive effect (p.111). He divided montage for categories :

Metric Montage

The basic criterion is the absolute length of the shots. The shots are joined together according to their lengths in formula-scheme. This is realized in the repetition of these formulas.(Eisenstein, 2009e, p.116)
Mechanical acceleration increases a tension. Lengths of shots are shorter and repeated.

Rhythmic Montage

is a special variant of metric montage. In this category the length of the shot depends of the content. The content determines the length. The lengths is flexible. As an classic sample Eisenstein give us The 'Odessa Steps':

He explains that the 'drumbeat' of the soldiers' feet descending the steps destroys all metrical conventions. It occurs outside the intervals prescribed by the metre and each time it appears in a different shot resolution. The final build-up of tension is produced by swiching from the rhythm of the solders' tread as they descend the steps to another, new form of movement- the next stage in the intensification of the same action – the pram rolling down the steps. Here the pram works in relation to the feet as a direct staged accelerator. The 'descent' of the feet becomes the 'rolling down' of the pram. (Eisenstein, 2009e, p.117)

Tonal Montage

This is about the next stage after rhythmic montage. In this category the concept of movement includes all sort of vibration that come from the shot. It is characterized by the emotional tonality of the shot i.e. by an apparently 'impressionistic' measurement, it is a simple delusion (Eisenstein 2009e, p. 118). This characteristic of the shot can be measured precisely. It is constructed on the dominant emotional resonance of the shot. If we designate the shot as a more gloomy, we can play with lighting and use specific degree of the illumination to get this effect. It is light tonality. As an example Eisenstein put the fog in the Odessa Port sequence in Battleship Potemkin.

He explained that in this sequence montage is built exclusively on the emotional resonance of individual shots, on the rhythmic vibrations that do not produce spatial transposition. In this regard it is interesting that, alongside the basic tonal dominant, a second accessory rhythmic dominant of shots is operating in the same way. It acts as a link between the tonal construction of this particular scene and the rhythmic tradition, whose furthest development is tonal montage as a whole. (Eisenstein, 2009e, p. 118).

Overtonal Montage

It is the furthest development of tonal montage. We can say that it is accumulation of all categories of montage. All stimulants in the shot. Eisenstein explained the way of evolution as category of montage. He claimed that transition from the metric to the rhythmic method arose from the emergency of conflict between the length of the shot and movement within the shot. The transition to tonal montage result from the conflict between the rhythmic and tonal principles of the shot. Lastly, overtonal montage resulted from the conflict between the tonal principle of the shot (the dominant) and the overtonal. (Eisenstein, 2009e, p. 120)

As a highest category of montage he established Intellectual Montage as a montage of overtones of an intellectual order. It is about conflict between intellectual effects. Combination of shots give us an abstract image and there is no need to explain it. As an example we can use the sequence of the gods in October.